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Exercises Sequences, counting and probability

Example 5 from Algebra and Trigonometry, 13.1 Sequences and Their Notations

Writing the Terms of a Sequence Defined by a Recursive Formula

Write the first five terms of the sequence defined by the recursive formula.

\begin{array}{l} \begin{array}{l} a_{1} = 9 \end{array} \\ a_{n} = 3 a_{n - 1} - 20, for n \geq 2 \end{array}

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 first term is given in the formula. For each subsequent term, we replace $a_{n - 1}$ with the value of the preceding term.
$\begin{array}{l} n = 1 \begin{array}{l} \end{array} & a_{1} = 9 \\ n = 2 & a_{2} = 3 a_{1} - 20 = 3 (9) - 20 = 27 - 20 = 7 \\ n = 3 & a_{3} = 3 a_{2} - 20 = 3 (7) - 20 = 21 - 20 = 1 \\ n = 4 & a_{4} = 3 a_{3} - 20 = 3 (1) - 20 = 3 - 20 = - 17 \\ n = 5 & a_{5} = 3 a_{4} - 20 = 3 (- 17) - 20 = - 51 - 20 = - 71 \end{array}$
The first five terms are ${9, 7,} .$ See Figure 5.

Figure 5

Your turn

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

Write the first five terms of the sequence defined by the recursive formula.

\begin{array}{l} a_{1} = 2 \\ a_{n} = 2 a_{n - 1} + 1, for n \geq 2 \end{array}

How did it go?

PreviousWriting an Explicit Formula for the nth Term of a Sequence Next Writing the Terms of a Sequence Defined by a Recursive Formula

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 13.1, 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.