Question

Use the initial term and the recursive formula to find an explicit formula for the sequence an. Write your answer in simplest form. an=-2 an = an-1 -13 an=​

Answer 1

From the recursive rule

$a_n = a_{n-1}-13$

it follows that

$a_{n-1}=a_{n-2}-13 \implies a_n = a_{n-2} - 2\times13$

$a_{n-2}=a_{n-3}-13 \implies a_n = a_{n-3} - 3\times13$

and so on. Notice how the subscript on a on the right side and the coefficient multiplied by 13 add up to n (n - 2 + 2 = n; n - 3 + 3 = n; and so on). If we continue the pattern, we'll end up with

$a_n = a_1 + (n-1)\times13$

so that the explicit rule for the n-th term in the sequence is

$a_n = -2 + 13(n-1) \implies a_n = \boxed{13n-15}$