Question

Find the first term and the difference of an arithmetic progression if the sum of its first five even terms is equal to $15$ and the sum of the first three terms is equal to 3.

Answer 1

Sum of $1^{st}$ seven terms

$15=\frac{5}{2}(2a+(5-1\left)d\right)3=a+\frac{4d}{2}3=a+2d-\left(i\right)$

Sum of $1^{st}$ $3$ terms$=3$

So,$3a+3d=-3$ [Considering $1^{st}$ term as $a,a+d,a+2d$]

$a+d=-1-\left(ii\right)$

Solving $\left(i\right)$ & $\left(ii\right)$

$d=3-(-1)d=4$

Putting value of $d$ in $\left(ii\right)$

$a+4=-1a=-5$

So first term$=-5$ and difference $=4$