If the sum of $n$ terms of a series be $a + b n + c n^{2}$ , find the $n^{t h}$ term and the nature of the series.

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Solution

$S_{n} = a + b n + c n^{2}$

$T_{n} = S_{n} - S_{n - 1} = a + b n + c n^{2} - [a + b (n - a) + c (n - a)^{2}]$

$⟹ T_{n} = 2 c n + b - c = (b + c) + (n - 1) 2 c$

Therefore the series is an A.P with first term $b + c$ and common difference $2 c$

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If the sum of n terms of a series be a+bn+cn2, find the nth term and the nature of the series.

Sn=a+bn+cn2Tn=Sn−Sn−1=a+bn+cn2−[a+b(n−a)+c(n−a)2]⟹Tn=2cn+b−c=(b+c)+(n−1)2cTherefore the series is an A.P with first term b+c and common difference 2c

If the sum of $n$ terms of a series be $a + b n + c n^{2}$ , find the $n^{t h}$ term and the nature of the series.

$S_{n} = a + b n + c n^{2}$

$T_{n} = S_{n} - S_{n - 1} = a + b n + c n^{2} - [a + b (n - a) + c (n - a)^{2}]$

$⟹ T_{n} = 2 c n + b - c = (b + c) + (n - 1) 2 c$

Therefore the series is an A.P with first term $b + c$ and common difference $2 c$