{"id":698,"date":"2025-07-14T21:12:10","date_gmt":"2025-07-14T21:12:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=698"},"modified":"2026-01-15T19:06:04","modified_gmt":"2026-01-15T19:06:04","slug":"module-11-sequences-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-11-sequences-cheat-sheet\/","title":{"raw":"Sequences: Cheat Sheet","rendered":"Sequences: Cheat Sheet"},"content":{"raw":"
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Essential Concepts<\/h2>\r\n

Sequences and Their Notations<\/h3>\r\n

Sequence Basics<\/strong><\/p>\r\n\r\n

    \r\n \t
  • A sequence is a function whose domain is the set of positive integers (1, 2, 3...)<\/li>\r\n \t
  • Each number in the sequence is called a term<\/li>\r\n \t
  • Terms are denoted as [latex]a_1, a_2, a_3, \\ldots, a_n, \\ldots[\/latex] where [latex]a_1[\/latex] is the first term, [latex]a_2[\/latex] is the second term, and [latex]a_n[\/latex] is the [latex]n[\/latex]th term<\/li>\r\n \t
  • A finite sequence has a limited number of terms; an infinite sequence continues indefinitely<\/li>\r\n \t
  • The ellipsis ([latex]\\ldots[\/latex]) indicates the sequence continues<\/li>\r\n<\/ul>\r\n

    Explicit Formulas<\/strong><\/p>\r\n\r\n

      \r\n \t
    • An explicit formula defines the [latex]n[\/latex]th term of a sequence using the position [latex]n[\/latex]<\/li>\r\n \t
    • Allows you to find any term directly without finding previous terms<\/li>\r\n \t
    • Written in the form [latex]a_n = f(n)[\/latex]<\/li>\r\n \t
    • To find terms, substitute [latex]n = 1, 2, 3, \\ldots[\/latex] into the formula<\/li>\r\n<\/ul>\r\n

      Alternating Sequences<\/strong><\/p>\r\n\r\n

        \r\n \t
      • Sequences where terms alternate in sign (positive, negative, positive, negative, etc.)\r\n
          \r\n \t
        • Use [latex](-1)^n[\/latex] if the first term is negative<\/li>\r\n \t
        • Use [latex](-1)^{n-1}[\/latex] if the first term is positive<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n

          Recursive Formulas<\/strong><\/p>\r\n\r\n

            \r\n \t
          • A recursive formula defines each term using one or more preceding terms<\/li>\r\n \t
          • Must always state the initial term(s)<\/li>\r\n<\/ul>\r\n

            Arithmetic Sequences<\/h3>\r\n
              \r\n \t
            • An arithmetic sequence has a constant difference between consecutive terms<\/li>\r\n \t
            • The common difference [latex]d[\/latex] is found by subtracting any term from the next term: [latex]d = a_n - a_{n-1}[\/latex]<\/li>\r\n \t
            • Explicit Form: The [latex]n[\/latex]th term is given by [latex]a_n = a_1 + (n-1)d[\/latex]<\/li>\r\n \t
            • Recursive Form: Written as [latex]a_n = a_{n-1} + d[\/latex] for [latex]n \\geq 2[\/latex]. Requires stating the initial term [latex]a_1[\/latex]<\/li>\r\n<\/ul>\r\n

              Geometric Sequences<\/h3>\r\n
                \r\n \t
              • A geometric sequence has a constant ratio between consecutive terms<\/li>\r\n \t
              • The common ratio [latex]r[\/latex] is found by dividing any term by the previous term: [latex]r = \\frac{a_n}{a_{n-1}}[\/latex]<\/li>\r\n \t
              • Explicit Form: The [latex]n[\/latex]th term is given by [latex]a_n = a_1 r^{n-1}[\/latex]<\/li>\r\n \t
              • Recursive Form: Written as [latex]a_n = r \\cdot a_{n-1}[\/latex] for [latex]n \\geq 2[\/latex]<\/li>\r\n<\/ul>\r\n

                Series and Summation Notation<\/h3>\r\n
                  \r\n \t
                • A series is the sum of the terms in a sequence<\/li>\r\n \t
                • Summation notation uses the Greek letter sigma ([latex]\\Sigma[\/latex]) to represent sums<\/li>\r\n \t
                • Format: [latex]\\sum_{k=1}^{n} a_k[\/latex] means sum [latex]a_k[\/latex] from [latex]k=1[\/latex] to [latex]k=n[\/latex]<\/li>\r\n \t
                • [latex]k[\/latex] is the index of summation<\/li>\r\n \t
                • The lower limit is the starting value; the upper limit is the ending value<\/li>\r\n<\/ul>\r\n

                  Arithmetic Series<\/strong><\/p>\r\n\r\n

                    \r\n \t
                  • The sum of the terms of an arithmetic sequence<\/li>\r\n \t
                  • Formula for the sum of the first [latex]n[\/latex] terms: [latex]S_n = \\frac{n}{2}(a_1 + a_n)[\/latex]<\/li>\r\n<\/ul>\r\n

                    Geometric Series<\/strong><\/p>\r\n\r\n

                      \r\n \t
                    • The sum of the terms of a geometric sequence<\/li>\r\n \t
                    • Formula for the sum of the first [latex]n[\/latex] terms: [latex]S_n = \\frac{a_1(1-r^n)}{1-r}[\/latex] where [latex]r \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n

                      Infinite Geometric Series<\/strong><\/p>\r\n\r\n

                        \r\n \t
                      • The sum exists only when [latex]-1 < r < 1[\/latex]<\/li>\r\n \t
                      • Formula for the sum: [latex]S = \\frac{a_1}{1-r}[\/latex] when [latex]-1 < r < 1[\/latex]<\/li>\r\n \t
                      • If [latex]|r| \\geq 1[\/latex], the series diverges (sum does not exist)<\/li>\r\n<\/ul>\r\n

                        Key Equations<\/h2>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                        [latex]n[\/latex]th term of an arithmetic sequence<\/strong><\/td>\r\n[latex]a_n = a_1 + (n-1)d[\/latex]<\/td>\r\n<\/tr>\r\n
                        Recursive formula for arithmetic sequence<\/strong><\/td>\r\n[latex]a_n = a_{n-1} + d, \\, n \\geq 2[\/latex]<\/td>\r\n<\/tr>\r\n
                        [latex]n[\/latex]th term of a geometric sequence<\/strong><\/td>\r\n[latex]a_n = a_1 r^{n-1}[\/latex]<\/td>\r\n<\/tr>\r\n
                        Recursive formula for geometric sequence<\/strong><\/td>\r\n[latex]a_n = r \\cdot a_{n-1}, \\, n \\geq 2[\/latex]<\/td>\r\n<\/tr>\r\n
                        Sum of first [latex]n[\/latex] terms of arithmetic series<\/strong><\/td>\r\n[latex]S_n = \\frac{n}{2}(a_1 + a_n)[\/latex]<\/td>\r\n<\/tr>\r\n
                        Sum of first [latex]n[\/latex] terms of geometric series<\/strong><\/td>\r\n[latex]S_n = \\frac{a_1(1-r^n)}{1-r}, \\, r \\neq 1[\/latex]<\/td>\r\n<\/tr>\r\n
                        Sum of infinite geometric series<\/strong><\/td>\r\n[latex]S = \\frac{a_1}{1-r}, \\, -1 < r < 1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                        Glossary<\/h2>\r\n

                        arithmetic sequence<\/strong><\/p>\r\n

                        A sequence where the difference between consecutive terms is always the same (constant difference [latex]d[\/latex]).<\/p>\r\n

                        common difference<\/strong><\/p>\r\n

                        The constant value [latex]d[\/latex] added to each term to get the next term in an arithmetic sequence, calculated as [latex]d = a_n - a_{n-1}[\/latex].<\/p>\r\n

                        common ratio<\/strong><\/p>\r\n

                        The constant value [latex]r[\/latex] by which each term is multiplied to get the next term in a geometric sequence, calculated as [latex]r = \\frac{a_n}{a_{n-1}}[\/latex].<\/p>\r\n

                        diverges<\/strong><\/p>\r\n

                        A series diverges when its sum is not a real number or does not approach a finite value.<\/p>\r\n

                        ellipsis<\/strong><\/p>\r\n

                        The symbol ([latex]\\ldots[\/latex]) used to indicate that a sequence or pattern continues indefinitely.<\/p>\r\n

                        explicit formula<\/strong><\/p>\r\n

                        A formula that defines the [latex]n[\/latex]th term of a sequence using the position [latex]n[\/latex] in the sequence.<\/p>\r\n

                        finite sequence<\/strong><\/p>\r\n

                        A sequence with a limited number of terms.<\/p>\r\n

                        geometric sequence<\/strong><\/p>\r\n

                        A sequence where the ratio between consecutive terms is always the same (constant ratio [latex]r[\/latex]).<\/p>\r\n

                        index of summation<\/strong><\/p>\r\n

                        The variable (often [latex]k[\/latex], [latex]i[\/latex], or [latex]n[\/latex]) used in summation notation to represent the position of terms being summed.<\/p>\r\n

                        infinite sequence<\/strong><\/p>\r\n

                        A sequence that continues indefinitely without ending.<\/p>\r\n

                        infinite series<\/strong><\/p>\r\n

                        The sum of all terms in an infinite sequence.<\/p>\r\n

                        lower limit of summation<\/strong><\/p>\r\n

                        The starting value of the index in summation notation.<\/p>\r\n

                        [latex]n[\/latex]th term of the sequence<\/strong><\/p>\r\n

                        The general term of a sequence, denoted [latex]a_n[\/latex], representing any term based on its position [latex]n[\/latex].<\/p>\r\n

                        partial sum<\/strong><\/p>\r\n

                        The sum of the first [latex]n[\/latex] terms of a series, denoted [latex]S_n[\/latex].<\/p>\r\n

                        recursive formula<\/strong><\/p>\r\n

                        A formula that defines each term of a sequence using one or more preceding terms. Must include initial term(s).<\/p>\r\n

                        sequence<\/strong><\/p>\r\n

                        A function whose domain is the set of positive integers; an ordered list of numbers following a specific pattern.<\/p>\r\n

                        series<\/strong><\/p>\r\n

                        The sum of the terms in a sequence.<\/p>\r\n

                        sigma<\/strong><\/p>\r\n

                        The Greek letter [latex]\\Sigma[\/latex] used in summation notation to indicate a sum.<\/p>\r\n

                        summation notation<\/strong><\/p>\r\n

                        A notation using the sigma symbol ([latex]\\Sigma[\/latex]) to represent the sum of terms in a series.<\/p>\r\n

                        term<\/strong><\/p>\r\n

                        An individual number in a sequence.<\/p>\r\n

                        upper limit of summation<\/strong><\/p>\r\n

                        The ending value of the index in summation notation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

                        \n
                        \n
                        \n
                        \n
                        \n

                        Essential Concepts<\/h2>\n

                        Sequences and Their Notations<\/h3>\n

                        Sequence Basics<\/strong><\/p>\n

                          \n
                        • A sequence is a function whose domain is the set of positive integers (1, 2, 3…)<\/li>\n
                        • Each number in the sequence is called a term<\/li>\n
                        • Terms are denoted as [latex]a_1, a_2, a_3, \\ldots, a_n, \\ldots[\/latex] where [latex]a_1[\/latex] is the first term, [latex]a_2[\/latex] is the second term, and [latex]a_n[\/latex] is the [latex]n[\/latex]th term<\/li>\n
                        • A finite sequence has a limited number of terms; an infinite sequence continues indefinitely<\/li>\n
                        • The ellipsis ([latex]\\ldots[\/latex]) indicates the sequence continues<\/li>\n<\/ul>\n

                          Explicit Formulas<\/strong><\/p>\n

                            \n
                          • An explicit formula defines the [latex]n[\/latex]th term of a sequence using the position [latex]n[\/latex]<\/li>\n
                          • Allows you to find any term directly without finding previous terms<\/li>\n
                          • Written in the form [latex]a_n = f(n)[\/latex]<\/li>\n
                          • To find terms, substitute [latex]n = 1, 2, 3, \\ldots[\/latex] into the formula<\/li>\n<\/ul>\n

                            Alternating Sequences<\/strong><\/p>\n

                              \n
                            • Sequences where terms alternate in sign (positive, negative, positive, negative, etc.)\n
                                \n
                              • Use [latex](-1)^n[\/latex] if the first term is negative<\/li>\n
                              • Use [latex](-1)^{n-1}[\/latex] if the first term is positive<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                Recursive Formulas<\/strong><\/p>\n

                                  \n
                                • A recursive formula defines each term using one or more preceding terms<\/li>\n
                                • Must always state the initial term(s)<\/li>\n<\/ul>\n

                                  Arithmetic Sequences<\/h3>\n
                                    \n
                                  • An arithmetic sequence has a constant difference between consecutive terms<\/li>\n
                                  • The common difference [latex]d[\/latex] is found by subtracting any term from the next term: [latex]d = a_n - a_{n-1}[\/latex]<\/li>\n
                                  • Explicit Form: The [latex]n[\/latex]th term is given by [latex]a_n = a_1 + (n-1)d[\/latex]<\/li>\n
                                  • Recursive Form: Written as [latex]a_n = a_{n-1} + d[\/latex] for [latex]n \\geq 2[\/latex]. Requires stating the initial term [latex]a_1[\/latex]<\/li>\n<\/ul>\n

                                    Geometric Sequences<\/h3>\n
                                      \n
                                    • A geometric sequence has a constant ratio between consecutive terms<\/li>\n
                                    • The common ratio [latex]r[\/latex] is found by dividing any term by the previous term: [latex]r = \\frac{a_n}{a_{n-1}}[\/latex]<\/li>\n
                                    • Explicit Form: The [latex]n[\/latex]th term is given by [latex]a_n = a_1 r^{n-1}[\/latex]<\/li>\n
                                    • Recursive Form: Written as [latex]a_n = r \\cdot a_{n-1}[\/latex] for [latex]n \\geq 2[\/latex]<\/li>\n<\/ul>\n

                                      Series and Summation Notation<\/h3>\n
                                        \n
                                      • A series is the sum of the terms in a sequence<\/li>\n
                                      • Summation notation uses the Greek letter sigma ([latex]\\Sigma[\/latex]) to represent sums<\/li>\n
                                      • Format: [latex]\\sum_{k=1}^{n} a_k[\/latex] means sum [latex]a_k[\/latex] from [latex]k=1[\/latex] to [latex]k=n[\/latex]<\/li>\n
                                      • [latex]k[\/latex] is the index of summation<\/li>\n
                                      • The lower limit is the starting value; the upper limit is the ending value<\/li>\n<\/ul>\n

                                        Arithmetic Series<\/strong><\/p>\n

                                          \n
                                        • The sum of the terms of an arithmetic sequence<\/li>\n
                                        • Formula for the sum of the first [latex]n[\/latex] terms: [latex]S_n = \\frac{n}{2}(a_1 + a_n)[\/latex]<\/li>\n<\/ul>\n

                                          Geometric Series<\/strong><\/p>\n

                                            \n
                                          • The sum of the terms of a geometric sequence<\/li>\n
                                          • Formula for the sum of the first [latex]n[\/latex] terms: [latex]S_n = \\frac{a_1(1-r^n)}{1-r}[\/latex] where [latex]r \\neq 1[\/latex]<\/li>\n<\/ul>\n

                                            Infinite Geometric Series<\/strong><\/p>\n

                                              \n
                                            • The sum exists only when [latex]-1 < r < 1[\/latex]<\/li>\n
                                            • Formula for the sum: [latex]S = \\frac{a_1}{1-r}[\/latex] when [latex]-1 < r < 1[\/latex]<\/li>\n
                                            • If [latex]|r| \\geq 1[\/latex], the series diverges (sum does not exist)<\/li>\n<\/ul>\n

                                              Key Equations<\/h2>\n\n\n\n\n\n\n\n\n\n
                                              [latex]n[\/latex]th term of an arithmetic sequence<\/strong><\/td>\n[latex]a_n = a_1 + (n-1)d[\/latex]<\/td>\n<\/tr>\n
                                              Recursive formula for arithmetic sequence<\/strong><\/td>\n[latex]a_n = a_{n-1} + d, \\, n \\geq 2[\/latex]<\/td>\n<\/tr>\n
                                              [latex]n[\/latex]th term of a geometric sequence<\/strong><\/td>\n[latex]a_n = a_1 r^{n-1}[\/latex]<\/td>\n<\/tr>\n
                                              Recursive formula for geometric sequence<\/strong><\/td>\n[latex]a_n = r \\cdot a_{n-1}, \\, n \\geq 2[\/latex]<\/td>\n<\/tr>\n
                                              Sum of first [latex]n[\/latex] terms of arithmetic series<\/strong><\/td>\n[latex]S_n = \\frac{n}{2}(a_1 + a_n)[\/latex]<\/td>\n<\/tr>\n
                                              Sum of first [latex]n[\/latex] terms of geometric series<\/strong><\/td>\n[latex]S_n = \\frac{a_1(1-r^n)}{1-r}, \\, r \\neq 1[\/latex]<\/td>\n<\/tr>\n
                                              Sum of infinite geometric series<\/strong><\/td>\n[latex]S = \\frac{a_1}{1-r}, \\, -1 < r < 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                              Glossary<\/h2>\n

                                              arithmetic sequence<\/strong><\/p>\n

                                              A sequence where the difference between consecutive terms is always the same (constant difference [latex]d[\/latex]).<\/p>\n

                                              common difference<\/strong><\/p>\n

                                              The constant value [latex]d[\/latex] added to each term to get the next term in an arithmetic sequence, calculated as [latex]d = a_n - a_{n-1}[\/latex].<\/p>\n

                                              common ratio<\/strong><\/p>\n

                                              The constant value [latex]r[\/latex] by which each term is multiplied to get the next term in a geometric sequence, calculated as [latex]r = \\frac{a_n}{a_{n-1}}[\/latex].<\/p>\n

                                              diverges<\/strong><\/p>\n

                                              A series diverges when its sum is not a real number or does not approach a finite value.<\/p>\n

                                              ellipsis<\/strong><\/p>\n

                                              The symbol ([latex]\\ldots[\/latex]) used to indicate that a sequence or pattern continues indefinitely.<\/p>\n

                                              explicit formula<\/strong><\/p>\n

                                              A formula that defines the [latex]n[\/latex]th term of a sequence using the position [latex]n[\/latex] in the sequence.<\/p>\n

                                              finite sequence<\/strong><\/p>\n

                                              A sequence with a limited number of terms.<\/p>\n

                                              geometric sequence<\/strong><\/p>\n

                                              A sequence where the ratio between consecutive terms is always the same (constant ratio [latex]r[\/latex]).<\/p>\n

                                              index of summation<\/strong><\/p>\n

                                              The variable (often [latex]k[\/latex], [latex]i[\/latex], or [latex]n[\/latex]) used in summation notation to represent the position of terms being summed.<\/p>\n

                                              infinite sequence<\/strong><\/p>\n

                                              A sequence that continues indefinitely without ending.<\/p>\n

                                              infinite series<\/strong><\/p>\n

                                              The sum of all terms in an infinite sequence.<\/p>\n

                                              lower limit of summation<\/strong><\/p>\n

                                              The starting value of the index in summation notation.<\/p>\n

                                              [latex]n[\/latex]th term of the sequence<\/strong><\/p>\n

                                              The general term of a sequence, denoted [latex]a_n[\/latex], representing any term based on its position [latex]n[\/latex].<\/p>\n

                                              partial sum<\/strong><\/p>\n

                                              The sum of the first [latex]n[\/latex] terms of a series, denoted [latex]S_n[\/latex].<\/p>\n

                                              recursive formula<\/strong><\/p>\n

                                              A formula that defines each term of a sequence using one or more preceding terms. Must include initial term(s).<\/p>\n

                                              sequence<\/strong><\/p>\n

                                              A function whose domain is the set of positive integers; an ordered list of numbers following a specific pattern.<\/p>\n

                                              series<\/strong><\/p>\n

                                              The sum of the terms in a sequence.<\/p>\n

                                              sigma<\/strong><\/p>\n

                                              The Greek letter [latex]\\Sigma[\/latex] used in summation notation to indicate a sum.<\/p>\n

                                              summation notation<\/strong><\/p>\n

                                              A notation using the sigma symbol ([latex]\\Sigma[\/latex]) to represent the sum of terms in a series.<\/p>\n

                                              term<\/strong><\/p>\n

                                              An individual number in a sequence.<\/p>\n

                                              upper limit of summation<\/strong><\/p>\n

                                              The ending value of the index in summation notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":156,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/698"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/698\/revisions"}],"predecessor-version":[{"id":5392,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/698\/revisions\/5392"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/156"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/698\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=698"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=698"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=698"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=698"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}