Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Find the Sum of the Series Whose Nth Term Is: 2n2 − 3n + 5 - Mathematics

Find the sum of the series whose nth term is:

2n2 − 3n + 5

#### Solution

Let $T_n$ be the nth term of the given series.
Thus, we have:

$T_n = 2 n^2 - 3n + 5$

Let $S_n$ be the sum of n terms of the given series.
Now,

$S_n = \sum^n_{k = 1} T_k$

$\Rightarrow S_n = \sum^n_{k = 1} \left( 2 k^2 - 3k + 5 \right)$

$\Rightarrow S_n = {2\sum}^n_{k = 1} k^2 - 3 \sum^n_{k = 1} k + \sum^n_{k = 1} 5$

$\Rightarrow S_n = \frac{2n\left( n + 1 \right)\left( 2n + 1 \right)}{6} - \frac{3n\left( n + 1 \right)}{2} + 5n$

$\Rightarrow S_n = \frac{2n\left( n + 1 \right)\left( 2n + 1 \right) - 9n\left( n + 1 \right) + 30n}{6}$

$\Rightarrow S_n = \frac{\left( 2 n^2 + 2n \right)\left( 2n + 1 \right) - 9 n^2 - 9n + 30n}{6}$

$\Rightarrow S_n = \frac{4 n^3 + 4 n^2 + 2 n^2 + 2n - 9 n^2 - 9n + 30n}{6}$

$\Rightarrow S_n = \frac{4 n^3 - 3 n^2 + 23n}{6}$

$\Rightarrow S_n = \frac{n\left( 4 n^2 - 3n + 23 \right)}{6}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 21 Some special series
Exercise 21.1 | Q 8.1 | Page 10

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