Find the sum of the following geometric series:
$(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + \dots$ to n terms;

Open in App

Solution

Let $S_{n} = (x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + \dots n t e r m s$ ;
Multiply and divide with (x-y) then we obtain
$S_{n} = \frac{(x - y) [(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + \dots n t e r m s}{(x - y)}$
$S_{n} = \frac{[(x - y) (x + y) + (x - y) (x^{2} + x y + y^{2}) + (x - y) (x^{3} + x^{2} y + x y^{2} + y^{3}) + \dots n t e r m s]}{(x - y)}$
$S_{n} = \frac{[(x^{2} - y^{2}) + (x^{3} - y^{3}) + (x^{4} - y^{4}) + \dots n t e r m s]}{(x - y)}$
$S_{n} = \frac{(x^{2} + x^{3} + x^{4} + \dots n t e r m s) - y^{2} + y^{3} + y^{4} \dots n t e r m s}{(x - y)}$
$S_{n} = \frac{[x^{2} (1 + x + x^{2} + \dots n t e r m s) - y^{2} (1 + y + y^{2} + \dots n t e r m s)]}{(x - y)}$
$S_{n} = ⎡ ⎢ ⎣ \frac{{\frac{x^{2} (1 - x^{n})}{1 - x}} - {\frac{y^{2} (1 - y^{n})}{(1 - y)}}}{(x - y)} ⎤ ⎥ ⎦$

Suggest Corrections

Similar questions

(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + {x y}^{2} + y^{3}) + . . .

n

(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + . . . n

=

If $| x | < 1$ and $| y | < 1,$ find the sum to infinity of the series

$(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + . . . .$

Find the sum to n terms of the series $\frac{1}{1.3} + \frac{1}{3.5} + \frac{1}{5.7} + . . . .$

| x | < 1, | y | < 1

x \neq y,

(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + . . . . . . \infty

∣ x ∣< 1

∣ y ∣< 1

(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + . . . . . \infty

Join BYJU'S Learning Program