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2 years ago

Please help with this

Mathematics

1 answer:

inysia [295]2 years ago

6 0

Answer:

$S_{n}$ = n² + 4n ; n = 11

Step-by-step explanation:

$S_{n}$ = $\frac{n}{2}$ [2 $a_{1}$ + (n - 1)d]

~~~~~~~~~

$a_{1}$ = 5

d = 7 - 5 = 2

$S_{n}$ = $\frac{n}{2}$ [ 2(5) + 2(n - 1)] = $\frac{n}{2}$ ( 8 + 2n ) = 4n + n²

$S_{n}$  = n² + 4n

n² + 4n = 165

n² + 4n - 165 = 0

(n + 15)(n - 11) = 0 ( n ≥ 0 ) ⇒ n = 11

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From Algebra we know that a second order polynomial is a perfect square if and only if $(x+y)^{2} = x^{2} + 2\cdot x\cdot y + y^{2}$ . From statement, we must fulfill the following identity:

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