Find the first term and the common difference of an arithmetic progression (A.P) if the sum of its first five even terms is equal to $15$ and the sum of the first three terms is equal to $-3$.
Let the first term of A.P $= t_1 = a$
Let the common difference of A.P $= d$
$n^{th}$ term of A.P $= t_n = a + \left(n - 1\right) d$
Given: Sum of the first five even terms of A.P $= 15$
i.e. $\;$ $t_2 + t_4 + t_6 + t_8 + t_{10} = 15$
i.e. $\;$ $\left(a + d\right) + \left(a + 3d\right) + \left(a + 5d\right) + \left(a + 7d\right) + \left(a + 9d\right) = 15$
i.e. $\;$ $5a + 25d = 15$
i.e. $\;$ $a + 5d = 3$ $\;\;\; \cdots \; (1)$
And, sum of the first three terms $= -3$
i.e. $\;$ $t_1 + t_2 + t_3 = -3$
i.e. $\;$ $a + \left(a + d\right) + \left(a + 2d\right) = -3$
i.e. $\;$ $3a + 3d = -3$
i.e. $\;$ $a + d = -1$ $\;\;\; \cdots \; (2)$
Subtracting equations $(1)$ and $(2)$ gives
$4d = 4$ $\implies$ $d = 1$
Substituting $d = 1$ in equation $(1)$ gives
$a + 5 = 3$ $\implies$ $a = -2$