The sum of the first n terms of the A.P. 2, 7 | vedclass

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Question

(A) Given the $A.P.$ is $2, 7, 12, 17, \ldots$Here,the first term $a = 2$ and the common difference $d = 7 - 2 = 5$.The sum of the first $n$ terms is

Vedclass · Accepted Answer

$20$

Vedclass · Answer

(A) Given the $A.P.$ is $2, 7, 12, 17, \ldots$Here,the first term $a = 2$ and the common difference $d = 7 - 2 = 5$.The sum of the first $n$ terms is given by the formula $S_n = \frac{n}{2}[2a + (n - 1)d]$.Substituting the given values: $990 = \frac{n}{2}[2(2) + (n - 1)5]$.$990 = \frac{n}{2}[4 + 5n - 5]$.$990 = \frac{n}{2}[5n - 1]$.$1980 = n(5n - 1)$.$5n^2 - n - 1980 = 0$.Solving the quadratic equation using the factorization method:$5n^2 - 100n + 99n - 1980 = 0$.$5n(n - 20) + 99(n - 20) = 0$.$(5n + 99)(n - 20) = 0$.This gives $n = 20$ or $n = -\frac{99}{5}$.Since $n$ must be a natural number,we discard the negative fraction.Therefore,$n = 20$.

The sum of the first $n$ terms of the $A.P.$ $2, 7, 12, 17, \ldots$ is $990$. Find the value of $n$.

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