{"id":1676,"date":"2025-07-25T19:12:23","date_gmt":"2025-07-25T19:12:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1676"},"modified":"2026-03-25T21:59:57","modified_gmt":"2026-03-25T21:59:57","slug":"geometric-sequences-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/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.\"\"The 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>

    [ohm_question hide_question_numbers=1]321967[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321968[\/ohm_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