QuestionFind a formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence $16, 7, -2, \ldots$ .

Studdy Solution

STEP 1

Assumptions1. The sequence given is $16,7,-,\ldots$ . The sequence is arithmetic, meaning the difference between consecutive terms is constant.

STEP 2

To find the explicit formula for an arithmetic sequence, we need to find the common difference, $d$ . This is the difference between any two consecutive terms. $d = a_{n} - a_{n-1}$

STEP 3

Let's calculate the common difference using the first two terms of the sequence.
$d =7 -16$

STEP 4

Calculate the common difference.
$d =7 -16 = -9$

STEP 5

Now that we have the common difference, we can write the explicit formula for the $n^{\text {th }}$ term of the sequence. The general formula for an arithmetic sequence is $a_{n} = a_{1} + (n-1) \cdot d$ where $a_{1}$ is the first term of the sequence, $n$ is the term number, and $d$ is the common difference.

STEP 6

Plug in the values for the first term and the common difference into the formula.
$a_{n} =16 + (n-1) \cdot -9$

STEP 7

implify the formula to get the explicit formula for the $n^{\text {th }}$ term of the sequence.
$a_{n} =16 -9(n-1)$ $a_{n} =16 -9n +9$ $a_{n} =25 -9n$ The explicit formula for the $n^{\text {th }}$ term of the sequence $16,7,-2, \ldots$ is $a_{n} =25 -9n$ .

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