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# 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?

Last updated date: 22nd Mar 2023
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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.

First term of an A.P$\left( {{a_1}} \right) = 17$
Last term of an A.P$\left( {{a_n}} \right) = 350$
Common difference$\left( d \right) = 9$
${a_n} = {a_1} + \left( {n - 1} \right)d$, (where n is number of terms)
$\Rightarrow 350 = 17 + \left( {n - 1} \right)9 \\ \Rightarrow \left( {n - 1} \right) = \dfrac{{350 - 17}}{9} = \dfrac{{333}}{9} = 37 \\ \Rightarrow n = 37 + 1 = 38 \\$
${{\text{S}}_n} = \dfrac{n}{2}\left( {{a_1} + {a_n}} \right) \\ \Rightarrow {{\text{S}}_n} = \dfrac{{38}}{2}\left( {17 + 350} \right) = \dfrac{{38}}{2}\left( {367} \right) = 6973 \\$