Question

If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.

विकल्प

• `(sqrt(ax))/(1 + sqrt(b)) + x/(1 + sqrt(y))`

• `sqrt(x)/(1 + sqrt(b)) + sqrt(x)/(1 + sqrt(y))`

• `sqrt(x)/(1 - sqrt(b)) + sqrt(x)/(1 - sqrt(y))`

• `(sqrt(ax))/(1 - sqrt(b)) + x/(1 - sqrt(y))`

Answer 1

If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is `underlinebb((sqrt(ax))/(1 - sqrt(b)) + x/(1 - sqrt(y))`.

Explanation:

We can break given series into two series as follows:

`(sqrt(ax) + sqrt(axb) + bsqrt(ax) + ......  ∞) + (x + xsqrt(y) + xy + ......  ∞)`

= `sqrt(ax)(1 + sqrt(b) + b + ....... ∞) + x(1 + sqrt(y) + y + .......  ∞)`

= `(sqrt(ax))/(1 - sqrt(b)) + x/(1 - sqrt(y))`