Answer:
[tex]S_{n}[/tex] = n² + 4n ; n = 11
Step-by-step explanation:
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [2 [tex]a_{1}[/tex] + (n - 1)d]
~~~~~~~~~
[tex]a_{1}[/tex] = 5
d = 7 - 5 = 2
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] [ 2(5) + 2(n - 1)] = [tex]\frac{n}{2}[/tex] ( 8 + 2n ) = 4n + n²
[tex]S_{n}[/tex] = n² + 4n
n² + 4n = 165
n² + 4n - 165 = 0
(n + 15)(n - 11) = 0 ( n ≥ 0 ) ⇒ n = 11
