# An arithmetic has 3 as it first term also the sum of the first 8 term is twice the sum of the 5 term. Find the common different?

Mar 6, 2017

$d = \frac{3}{4}$

#### Explanation:

We must write an equation using the formula ${s}_{n} = \frac{n}{2} \left(2 a + \left(n - 1\right) d\right)$, in order to solve for $d$, the common difference.

We know that ${s}_{8} = 2 \left({s}_{5}\right)$, and that $a = 3$, so our equation will be:

$\frac{8}{2} \left(2 \left(3\right) + \left(8 - 1\right) d\right) = 2 \left(\frac{5}{2} \left(2 \left(3\right) + \left(5 - 1\right) d\right)\right)$

$4 \left(6 + 7 d\right) = 5 \left(6 + 4 d\right)$

$24 + 28 d = 30 + 20 d$

$24 - 30 = 20 d - 28 d$

$- 6 = - 8 d$

$d = \frac{3}{4}$

If you recheck using the formula ${s}_{n} = \frac{n}{2} \left(2 a + \left(n - 1\right) d\right)$, you will find that ${s}_{5} = \frac{45}{2}$ and ${s}_{8} = 45$, which makes our answer correct (since $2 {s}_{5} = {s}_{8}$).

Hopefully this helps!