Find a formula for the n th term of the arithmetic sequence. $a _ { 5 } = 13 , a _ { 9 } = 3$
A) $a _ { n } = - \frac { 5 n - 51 } { 2 }$
B) $a _ { n } = \frac { 5 n - 51 } { 2 }$
C) $a _ { n } = 23 n - 10$
D) $\alpha _ { n } = 13 n + 10$
E) $a _ { n } = - \frac { 5 n + 51 } { 2 }$

Question

Quizplus · Accepted Answer

To find the formula for the nth term of an arithmetic sequence, we use the formula $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, and $d$ is the common difference. Given $a_5 = 13$ and $a_9 = 3$, we can find the common difference $d$ by subtracting these terms and dividing by the difference in their positions: $d = \frac{3 - 13}{9 - 5} = \frac{-10}{4} = -\frac{5}{2}$. To find $a_1$, we can use one of the given terms and solve for $a_1$: $13 = a_1 + 4(-\frac{5}{2})$, which gives $a_1 = 13 + 10 = 23$. Substituting $a_1$ and $d$ into the formula for $a_n$, we get $a_n = 23 + (n-1)(-\frac{5}{2})$, simplifying this gives $a_n = -\frac{5n - 51}{2}$, which matches choice A.

Find a Formula for the N Th Term of the Arithmetic