# Find the value of n given First term is a=-16, Common difference d=8, the sum of first n terms is 600?

Find the value of n given First term is a=-16, Common difference d=8, the sum of first n terms is 600?
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Jacey Humphrey
This is an Arithmetic sequence
The sum of the first n terms of this sequence is given by.
${S}_{n}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$
where a is the first term and d , the common difference.
here a = -16 , d = 8 and require to solve for n.
hence : $\frac{n}{2}\left[\left(2×-16\right)+8\left(n-1\right)\right]=600$
$\frac{n}{2}\left[-32+8n-8\right]=600⇒\frac{n}{2}\left(8n-40\right)=600$
distributing gives : $4{n}^{2}-20n-600=0$
Equated to zero since this is a quadratic equation.
$⇒4\left({n}^{2}-5n-150\right)=0$
$⇒4\left(n+10\right)\left(n-15\right)=0⇒n=-10\phantom{\rule{1ex}{0ex}}\text{or}\phantom{\rule{1ex}{0ex}}n=15$
but n > 0 hence n = 15