{"id":1674,"date":"2025-07-25T19:12:33","date_gmt":"2025-07-25T19:12:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1674"},"modified":"2026-03-25T22:05:20","modified_gmt":"2026-03-25T22:05:20","slug":"geometric-sequences-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/geometric-sequences-learn-it-4\/","title":{"raw":"Geometric Sequences: Learn It 4","rendered":"Geometric Sequences: Learn It 4"},"content":{"raw":"

Using Recursive Formulas for Geometric Sequences<\/h2>\r\nA recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. As with any recursive formula, the initial term must be given.\r\n\r\n
\r\n

recursive formula for a geometric sequence<\/h3>\r\nThe recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]a_1[\/latex] is\r\n

[latex]a_n = ra_{n-1}, n \\ge 2[\/latex]<\/p>\r\n\r\n<\/section>

Write a recursive formula for the following geometric sequence.\r\n

[latex]\\left\\{6,9,13.5,20.25,\\dots\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"34110\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"34110\"]\r\n\r\nThe first term is given as [latex]6[\/latex]. The common ratio can be found by dividing the second term by the first term.\r\n

[latex]r=\\dfrac{9}{6}=1.5[\/latex]<\/p>\r\nSubstitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[\/latex].\r\n

[latex]\\begin{align}&{a}_{n}=r\\cdot{a}_{n - 1} \\\\ &{a}_{n}=1.5\\cdot{a}_{n - 1}\\text{ for }n\\ge 2 \\\\ &{a}_{1}=6\\end{align}[\/latex]<\/p>\r\n\"Graph\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

The common ratio is the base of an exponential function!\r\n[latex]\\\\[\/latex]\r\n<\/strong>Using the definition of the geometric sequence\u00a0[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex], we have [latex]a_n=a_1\\left(r\\right)^{n-1}[\/latex].\r\n[latex]\\\\[\/latex]<\/strong>\r\nRecall the form of an exponential function [latex]f(x)=a\\left(b\\right)^x[\/latex].<\/section>
[ohm_question hide_question_numbers=1]321971[\/ohm_question]<\/section>
[ohm_question hide_question_numbers=1]321972[\/ohm_question]<\/section>","rendered":"

Using Recursive Formulas for Geometric Sequences<\/h2>\n

A recursive formula<\/strong> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. As with any recursive formula, the initial term must be given.<\/p>\n

\n

recursive formula for a geometric sequence<\/h3>\n

The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]a_1[\/latex] is<\/p>\n

[latex]a_n = ra_{n-1}, n \\ge 2[\/latex]<\/p>\n<\/section>\n

Write a recursive formula for the following geometric sequence.<\/p>\n

[latex]\\left\\{6,9,13.5,20.25,\\dots\\right\\}[\/latex]<\/p>\n