Question

Will give 25 points and brainliest answer if correct

Find an equation for the nth term of the arithmetic sequence.

a19 = -58, a21 = -164

Answer Choices

A) an = 896 - 53(n - 2)
B) an = 896 - 53(n - 1)
C) an = 896 + 53(n + 1)
D) an = 896 - 53(n + 1)

Answer 1

Let $d$ be the common difference between terms in the sequence $\{a_n\}_{n\ge1}$. Then

$a_{19}=-58$
$a_{20}=a_{19}+d$
$a_{21}=a_{19}+2d$
$\implies-164=-58+2d\implies d=-53$

The first term in the sequence, $a_1$, satisfies

$a_2=a_1+d$
$a_3=a_1+2d$
$a_4=a_1+3d$
...
$a_{19}=a_1+18d$
$\implies-58=a_1+18(-53)\implies a_1=896$

So the explicit formula for the $n$-th term in the sequence is

$a_n=a_1+(n-1)d\implies a_n=896-53(n-1)$