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Exercises Sequences, counting and probability

Example 4 from Algebra and Trigonometry, 13.3 Geometric Sequences

Writing Terms of Geometric Sequences Using the Explicit Formula

Given a geometric sequence with $a_{1} = 3$ and $a_{4} = 24,$ find $a_{2} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The sequence can be written in terms of the initial term and the common ratio $r .$

$3, 3 r, 3 r^{2}, 3 r^{3}, ...$

Find the common ratio using the given fourth term.

$\begin{array}{l} a_{n} = a_{1} r^{n - 1} \\ a_{4} = 3 r^{3} & Write the fourth term of sequence in terms of α_{1} and r \\ 24 = 3 r^{3} & Substitute 24 for a_{4} \\ 8 = r^{3} & Divide \\ r = 2 & Solve for the common ratio \end{array}$
Find the second term by multiplying the first term by the common ratio.

$\begin{array}{l} a_{2} & = 2 a_{1} \\ = 2 (3) \\ = 6 \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given a geometric sequence with $a_{2} = 4$ and $a_{3} = 32$ , find $a_{6} .$

How did it go?

PreviousUsing Recursive Formulas for Geometric Sequences Next Writing an Explicit Formula for the n th Term of a Geometric Sequence

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 13.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.