{"id":2717,"date":"2024-08-12T23:17:22","date_gmt":"2024-08-12T23:17:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2717"},"modified":"2025-08-15T17:16:21","modified_gmt":"2025-08-15T17:16:21","slug":"geometric-sequences-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/geometric-sequences-learn-it-2\/","title":{"raw":"Geometric Sequences: Learn It 2","rendered":"Geometric Sequences: Learn It 2"},"content":{"raw":"

Writing Terms of Geometric Sequences<\/h2>\r\nNow that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.\r\n\r\n
For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\r\n

[latex]\\begin{align}&{a}_{1}=-2 \\\\ &{a}_{2}=\\left(-2\\cdot 4\\right)=-8 \\\\ &{a}_{3}=\\left(-8\\cdot 4\\right)=-32 \\\\ &{a}_{4}=\\left(-32\\cdot 4\\right)=-128 \\end{align}[\/latex]<\/p>\r\nThe first four terms are [latex]\\left\\{-2,-8,-32,-128\\right\\}[\/latex].\r\n\r\n<\/section>

How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/strong>\r\n
    \r\n \t
  1. Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\r\n \t
  2. Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\r\n \t
  3. Write the terms separated by commons within brackets.<\/li>\r\n<\/ol>\r\n<\/section>
    Write the first five terms of the geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]r=-2[\/latex].\r\n[reveal-answer q=\"67806\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"67806\"]We start with the first term and multiply it by the common ratio. Then we multiply that result by the common ratio to get the next term, and so on.\r\n\r\n[caption id=\"attachment_2724\" align=\"aligncenter\" width=\"651\"]\"\" Writing terms of a geometric sequence[\/caption]\r\n\r\nThe sequence is [latex]3, -6, 12, -24, 48, \\dots[\/latex][\/hidden-answer]<\/section>
    List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=-2[\/latex].[reveal-answer q=\"650557\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"650557\"]Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}[\/latex]. Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex],\u00a0and so on.\r\n

    [latex]\\begin{align}&{a}_{1}=5 \\\\ &{a}_{2}=-2{a}_{1}=-10 \\\\ &{a}_{3}=-2{a}_{2}=20 \\\\ &{a}_{4}=-2{a}_{3}=-40 \\end{align}[\/latex]<\/p>\r\nThe first four terms are [latex]\\left\\{5,-10,20,-40\\right\\}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm2_question hide_question_numbers=1]24940[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]24941[\/ohm2_question]<\/section>","rendered":"

    Writing Terms of Geometric Sequences<\/h2>\n

    Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.<\/p>\n

    For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4[\/latex], we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.<\/p>\n

    [latex]\\begin{align}&{a}_{1}=-2 \\\\ &{a}_{2}=\\left(-2\\cdot 4\\right)=-8 \\\\ &{a}_{3}=\\left(-8\\cdot 4\\right)=-32 \\\\ &{a}_{4}=\\left(-32\\cdot 4\\right)=-128 \\end{align}[\/latex]<\/p>\n

    The first four terms are [latex]\\left\\{-2,-8,-32,-128\\right\\}[\/latex].<\/p>\n<\/section>\n

    How To: Given the first term and the common factor, find the first four terms of a geometric sequence.<\/strong><\/p>\n
      \n
    1. Multiply the initial term, [latex]{a}_{1}[\/latex], by the common ratio to find the next term, [latex]{a}_{2}[\/latex].<\/li>\n
    2. Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n
    3. Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/section>\n
      Write the first five terms of the geometric sequence with [latex]{a}_{1}=3[\/latex] and [latex]r=-2[\/latex].<\/p>\n