Find the sum of n terms of the series $1 + 3 x + 6 x^{2} + 10 x^{3} + 15 x^{4} + . . . .$

Open in App

Solution

$S = 1 + 3 x + 6 x^{2} + 10 x^{3} + 15 x^{4} + . . . . . . + \frac{n (n + 1)}{2} x^{n - 1}$
$x S = x + 3 x^{2} + 6 x^{3} + 10 x^{4} + . . . . . . + {\frac{(n - 1)}{2}}^{n} x^{n - 1} + \frac{n (n + 1)}{2} x^{n}$
$(1 - x) S = (1 + 2 x + 3 x^{2} + 4 x^{3} + 5 x^{4} + . . . . . . t o n t e r m s) - \frac{n (n + 1)}{2} x^{n}$
$S_{1} = 1 + 2 x + 3 x^{2} + . . . .$ up to n term
$S_{1} = \frac{1 - x^{n}}{{(1 - x)}^{2}} - \frac{n x^{n}}{(1 - x)}$
$∴ S = \frac{1 - x^{n}}{{(1 - x)}^{3}} - \frac{n x^{n}}{{(1 - x)}^{2}} - \frac{n (n + 1)}{2 (1 - x)}$

Suggest Corrections

Similar questions

1 + 3 x + 6 x^{2} + 10 x^{3} + . . . \infty

1 . The sum of the series 1 + 3x + 6 $x^{2}$ + 10 $x^{3}$ + ....... ∞ will be

n

1 + 2 x + 3 x^{2} + 4 x^{3} + . . .

1 + 3 x + 6 x^{2} + 10 x^{3} + . . . .

Join BYJU'S Learning Program