# How do you find the value of n given First term is a=-16, Common difference d=8, And the sum of first n terms is 600?

Feb 27, 2016

n = 15

#### Explanation:

This is an$\textcolor{b l u e}{\text{ Arithmetic sequence }}$

The sum of the first n terms of this sequence is given by.

${S}_{n} = \frac{n}{2} \left[2 a + \left(n - 1\right) d\right]$

where a is the first term and d , the common difference.

here a = -16 , d = 8 and require to solve for n.

hence : $\frac{n}{2} \left[\left(2 \times - 16\right) + 8 \left(n - 1\right)\right] = 600$

$\frac{n}{2} \left[- 32 + 8 n - 8\right] = 600 \Rightarrow \frac{n}{2} \left(8 n - 40\right) = 600$

distributing gives : $4 {n}^{2} - 20 n - 600 = 0$

Equated to zero since this is a quadratic equation.

$\Rightarrow 4 \left({n}^{2} - 5 n - 150\right) = 0$

$\Rightarrow 4 \left(n + 10\right) \left(n - 15\right) = 0 \Rightarrow n = - 10 \mathmr{and} n = 15$

but n > 0 hence n = 15