# How do you find the number of terms n given the sum s_n=182 of the series 50+42+34+26+18+...?

Aug 20, 2016

There are $7$ terms in this series.

#### Explanation:

We use the formula ${s}_{n} = \frac{n}{2} \left(2 a + \left(n - 1\right) d\right)$ to determine the sum of an arithmetic series.

Here, $d = - 8$, n = ?, $a = 50$ and ${s}_{n} = 182$. We will therefore be solving for $n$.

$182 = \frac{n}{2} \left(2 \left(50\right) + \left(n - 1\right) - 8\right)$

$182 = \frac{n}{2} \left(100 - 8 n + 8\right)$

$182 = \frac{n}{2} \left(108 - 8 n\right)$

$182 = \frac{108 n}{2} - \frac{8 {n}^{2}}{2}$

$182 = 54 n - 4 {n}^{2}$

$4 {n}^{2} - 54 n + 182 = 0$

$n = \frac{- \left(- 54\right) \pm \sqrt{{54}^{2} - 4 \times 4 \times 182}}{2 \times 4}$

$n = \frac{54 \pm \sqrt{4}}{8}$

$n = \frac{54 + 2}{8} \mathmr{and} n = \frac{54 - 2}{8}$

$n = 7 \mathmr{and} n = 6.5$

Since a number of terms that isn't a whole number isn't possible, the aforementioned series has $7$ terms.

Hopefully this helps!