Question

# The first term 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?

Hint: The sum of the A.P is ${{\text{S}}_n} = \dfrac{n}{2}\left( {{a_1} + {a_n}} \right)$, use this property of A.P. Where ${a_1}$and ${a_n}$ are the first and last term respectively, and n is the number of terms.

Given: -
First term of an A.P$\left( {{a_1}} \right) = 17$
Last term of an A.P$\left( {{a_n}} \right) = 350$
Common difference$\left( d \right) = 9$
Now, we have to find out the number of terms and the sum of this A.P.
First find out the number of terms of this A.P.
${a_n} = {a_1} + \left( {n - 1} \right)d$, (where n is number of terms)
$\Rightarrow 350 = 17 + \left( {n - 1} \right)9 \\ \Rightarrow \left( {n - 1} \right) = \dfrac{{350 - 17}}{9} = \dfrac{{333}}{9} = 37 \\ \Rightarrow n = 37 + 1 = 38 \\$
As we know the sum of the A.P is given as
${{\text{S}}_n} = \dfrac{n}{2}\left( {{a_1} + {a_n}} \right) \\ \Rightarrow {{\text{S}}_n} = \dfrac{{38}}{2}\left( {17 + 350} \right) = \dfrac{{38}}{2}\left( {367} \right) = 6973 \\$
So, the required number of terms in an A.P is 38 and the required sum is 6973.

Note: - In such types of questions the key concept we have to remember is that always remember the general formula to find out the number of terms and the sum of an A.P which is stated above, so first find out the number of terms, then find out the required sum of the series.