The sum of n terms of an AP is $3 n^{2} + 5 n$ , find the AP. Hence find its $16^{t h}$ term.

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Solution

$S_{n} = 3 n^{2} + 5 n \Rightarrow a_{n} = 5 n - 5 n - 1 = 3 n^{2} + 5 n - 3 {(n - 1)}^{2} + 5 (n - 1) \Rightarrow a_{n} = 3 n^{2} + 5 n - 3 n^{2} + 6 n - 3 + 5 n - 5 \Rightarrow a_{n} = 16 n - 8 ∴ a_{1} = 16 \times 1 - 8 \Rightarrow a_{1} = 8 \Rightarrow a_{2} = 16 \times 2 - 8 \Rightarrow a_{2} = 32 - 8 = 24 \Rightarrow a_{3} = 16 \times 3 - 8 \Rightarrow a_{3} = 48 - 8 = 40 8, 24, 40 F i r s t t e r m = 8 c o m m o n d i f f e r e n c e = 24 - 8 = 16 \Rightarrow T_{16} = a + (16 - 1) d \Rightarrow T_{16} = 8 + 15 \times 16 = 8 + 240 = 248$

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The sum of n terms of an AP is 3n2+5n, find the AP. Hence find its 16th term.

Sn=3n2+5n⇒an=5n−5n−1=3n2+5n−3(n−1)2+5(n−1)⇒an=3n2+5n−3n2+6n−3+5n−5⇒an=16n−8∴a1=16×1−8⇒a1=8⇒a2=16×2−8⇒a2=32−8=24⇒a3=16×3−8⇒a3=48−8=408,24,40Firstterm=8commondifference=24−8=16⇒T16=a+(16−1)d⇒T16=8+15×16=8+240=248

The sum of n terms of an AP is $3 n^{2} + 5 n$ , find the AP. Hence find its $16^{t h}$ term.