Question
In an $A P$, if $S_n=n(4 n+1)$, find the $A P$.
✓
Answer
We know that, the $n^{\text {th }}$ term of an AP is
$a_n = S_n - S_{n-1}$[$\because$ $S_n = n(4n + 1)]$
$a_n = n(4n + 1) - (n - 1){4(n - 1) + 1}$
$a_n = 4n^2 + n - (n - 1)(4n - 3)$
$a_n = 4n^2 + n - 4n^2 + 3n + 4n - 3$
$a_n = 8n - 3$
Put n $= 1, a_1 = 8(1) - 3 = 5$
Put n $= 2, a_2 = 8(2) - 3 = 16 - 3 = 13$
Put n$ = 3, a_3 = 8(3) - 3 = 24 - 3 = 21$
Hence, the required $AP$ is $5, 13, 21, ......$
$a_n = S_n - S_{n-1}$[$\because$ $S_n = n(4n + 1)]$
$a_n = n(4n + 1) - (n - 1){4(n - 1) + 1}$
$a_n = 4n^2 + n - (n - 1)(4n - 3)$
$a_n = 4n^2 + n - 4n^2 + 3n + 4n - 3$
$a_n = 8n - 3$
Put n $= 1, a_1 = 8(1) - 3 = 5$
Put n $= 2, a_2 = 8(2) - 3 = 16 - 3 = 13$
Put n$ = 3, a_3 = 8(3) - 3 = 24 - 3 = 21$
Hence, the required $AP$ is $5, 13, 21, ......$
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