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  4. A sequence is such that the sum of its any number of terms, beginning from the first, is four times as large as the square of the number of terms. If the n text th term of such a sequence is 996 , then value of n is equal to

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Q. A sequence is such that the sum of its any number of terms, beginning from the first, is four times as large as the square of the number of terms. If the $n ^{\text {th }}$ term of such a sequence is 996 , then value of $n$ is equal to

Sequences and Series

Solution:

Given that $S _{ n }=4 n ^2$
$\therefore T _{ n }= S _{ n }- S _{ n -1}=4 n ^2-4( n -1)^2=4\left[ n ^2-\left( n ^2+1-2 n \right)\right]=4(2 n -1)=8 n -4$ Hence $T _1, T _2, T _3, T _4, \ldots \ldots \ldots . . .$. are in A.P.
i.e., $ 4,12,20,28, \ldots \ldots .$.
Now, $T _{ n }=996=8 n -4 \Rightarrow n =125$.