Answer to Question #117817 in Algebra for james urio

Question

The sum of the first five terms of an AP is 35. The sum of the next five terms of this is AP is 85. Find the first term
and the common difference

Answer 1

If $a$ is the first term of an AP and $d$ is common difference, then sum of first $n$ terms is $an+\frac{n(n-1)}{2}d$.

Since sum of first five terms is $35$ and sum of next five terms is $85$, we obtain that sum of first ten terms is $120$.

So $35=5a+\frac{5(5-1)}{2}d=5a+10d$ and $120=10a+\frac{10(10-1)}{2}d=10a+45d$.

We have $120-35\cdot 2=(10a+45d)-2(5a+10d)$, that is $50=25d$, so $d=2$.

Then we have $35=5a+10d=5a+10\cdot 2$, so $a=3$

Answer: the first term is $3$, the common difference is $2$