Sequences and series help. Finding the first term and the common difference.

Question

The question states: In an AP, The sum of the second and fourth term is $15$. The sum of the fifth term and the sixth term is $25$. Find the first term and the common difference.

Thank you in advance to anybody who helps :)

Answer 1

Hint:

For an arithmetic sequence,

$$a_n=a_1+(n-1)d$$

So the fact that $a_2+a_4=15$ says that $2a_1+4d=15$. Now use the other statement similarly. Then you have two linear equations with the two variables $a_1$, $d$.

Answer 2

For an arithmetic progression, we have (as our first 6 terms):

$$n, n + ad, n + 2ad, n + 3ad, ..., n + 5ad$$

So

$$ (n + d) + (n + 3d) = 15$$ $$ (n + 4d) + (n + 5d) = 25$$

This is a system of two equations in two variables, which means you can solve it to find the value of $n$ and $d$.