Arithmetic Progression

The first and the last term of an A.P are $17$ and $350$ respectively. If the common difference is $9$, how many terms are there and what is their sum?


First term of A.P $= a = 17$

Common difference of A.P $= d = 9$

Let the $n^{th}$ term be the last term.

Last term of A.P $= t_n = a + \left(n - 1\right)d = 350$ $\;\;\; \cdots \; (1)$

Substituting the values of $a$ and $d$ in equation $(1)$ we have,

$17 + \left(n - 1\right) \times 9 = 350$

i.e. $\;$ $n - 1 = \dfrac{333}{9}$

i.e. $\;$ $n = \dfrac{333}{9} + 1 = \dfrac{342}{9} = 38$

$\therefore \;$ Number of terms in A.P $= n = 38$

Sum of $n$ terms of A.P $= S_n = \dfrac{n}{2} \left[2a + \left(n - 1\right)d\right]$

$\therefore \;$ $S_{38} = \dfrac{38}{2} \left[2 \times 17 + \left(38 - 1\right) \times 9\right]$

i.e. $\;$ $S_{38} = 19 \left(34 + 333\right) = 6973$