The sum of the first n terms of an A.P. is gi | vedclass

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(A) The $n^{th}$ term of an $A.P.$ is given by the formula $T_{n} = S_{n} - S_{n-1}$ for $n > 1$.Given $S_{n} = 7n^{2} - 3n$.Then $S_{n-1} = 7(n-1)^{2

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$14n - 10$

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(A) The $n^{th}$ term of an $A.P.$ is given by the formula $T_{n} = S_{n} - S_{n-1}$ for $n > 1$.Given $S_{n} = 7n^{2} - 3n$.Then $S_{n-1} = 7(n-1)^{2} - 3(n-1) = 7(n^{2} - 2n + 1) - 3n + 3 = 7n^{2} - 14n + 7 - 3n + 3 = 7n^{2} - 17n + 10$.Now,$T_{n} = (7n^{2} - 3n) - (7n^{2} - 17n + 10) = 7n^{2} - 3n - 7n^{2} + 17n - 10 = 14n - 10$.For $n = 1$,$T_{1} = S_{1} = 7(1)^{2} - 3(1) = 7 - 3 = 4$.Checking the formula $14n - 10$ for $n = 1$,we get $14(1) - 10 = 4$,which matches $T_{1}$.Thus,the $n^{th}$ term is $T_{n} = 14n - 10$ for all $n \geq 1$.

The sum of the first $n$ terms of an $A.P.$ is given by $S_{n} = 7n^{2} - 3n$. Find the $n^{th}$ term of the $A.P.$

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