Find the sum of n terms of the series $2.2 + 4.4 + 7.8 + 11.16 + 16.32 + . . . .$

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Solution

$\Rightarrow 24711 16....$
$234 5....$
$11 1....$
$∴ u_{n} = 2 + 2 (n - 1) + \frac{(n - 1) (n - 2)}{2}$
$= \frac{4 n + n^{2} - 3 n + 2}{2} = \frac{n^{2} + n + 2}{2}$
$∴ t_{n} = (\frac{n^{2} + n + 2}{2}) \cdot 2^{n}$
$= (n^{2} + n + 2) 2^{n - 1}$
Let $t_{n} = (A n^{2} + B n + C) 2^{n - 1} - (A {(n - 1)}^{2} + B (n - 1) + C) 2^{n - 2}$
Putting $n = 1, 2, 3,$ we get;-
$A = 2, B = - 2, C = 8$
$t_{n} = (2 n^{2} + 2 n + 8) \cdot 2^{n - 1} - (2 {(n - 1)}^{2} + 2 (n - 1) + 8) 2^{n - 2}$
$t_{n - 1} = (2 {(n - 1)}^{2} + 2 (n - 1) + 8) 2^{n - 2} - (2 {(n - 2)}^{2} + 2 (n - 2) + 8) 2^{n - 3}$
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$t_{1} = 14 - \frac{8}{2}$

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