The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + ... is [2023]

Question

The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + ... is [2023]

Smart Achievers · Accepted Answer

(3)

$Let S = 5 + 11 + 19 + 29 + \dots + T_{n}$

$\Rightarrow S = 5 + 11 + 19 + \dots + T_{n - 1} + T_{n}$

$Subtracting above two equations, we get:$

$0 = 5 + {6 + 8 + 10 + 12 + \dots + (n - 1) terms} - T_{n}$

$\Rightarrow T_{n} = 5 + 2 (3 + 4 + 5 + \dots upto (n - 1) terms)$

$= 5 + 2 (1 + 2 + 3 + \dots upto (n + 1) terms - 1 - 2)$

$\Rightarrow T_{n} = 5 + 2 \cdot \frac{(n + 1) (n + 2)}{2} - 6 = n^{2} + 3 n + 2 - 1$

$\Rightarrow T_{n} = n^{2} + 3 n + 1$ $∴$ $S_{n} = \sum n^{2} + 3 \sum n + \sum 1$

$\Rightarrow \frac{n (n + 1) (2 n + 1)}{6} + \frac{3 (n + 1) n}{2} + n$

$S_{20} = \frac{20 \times 21 \times 41}{6} + \frac{3 \times 20 \times 21}{2} + 20 = 3520$

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