In an arithmetic progression, the third term is $8$ and the seventh term is $12$. Find the
- first term;
- common difference;
- sum of first $20$ terms.
Let
first term of the A.P be $= t_1 = a$
common difference $= d$
$n^{th}$ term of AP $= t_n = a + \left(n - 1\right)d$
$\therefore \;$ third term of AP $= t_3 = a + 2d = 8$ $\;\;\;$ [given] $\;\;\; \cdots \; (1)$
seventh term of AP $= t_7 = a + 6d = 12$ $\;\;\;$ [given] $\;\;\; \cdots \; (2)$
Solving equations $(1)$ and $(2)$ simultaneously gives
$4d = 4$ $\implies$ $d = 1$
Substituting the value of $d$ in equation $(1)$ gives
$a = 8 - 2d = 8 - 2 = 6$
$\therefore \;$ First term $= a = 6$; $\;$ common difference $= d = 1$
Sum of $n$ terms of AP $= S_n = \dfrac{n}{2} \left[2a + \left(n - 1\right)d\right]$
$\therefore \;$ Sum of first $20$ terms $= S_{20} = \dfrac{20}{2} \left[2 \times 6 + \left(20 - 1\right) \times 1\right]$
i.e. $S_{20} = 10 \left[12 + 19\right] = 310$