jmc

Question

Let a and b be real numbers such that ab + ab^2 + ab^3 + = 4.Find aa + b + a(a + b)^2 + a(a + b)^3 + .

Accepted Answer

From the formula for an infinite geometric series,

\frac{a / b}{1 - 1/ b} = 4.

Then

\frac{a}{b - 1} = 4,

so

a = 4 (b - 1) .

Again from the formula,

\frac{a}{a + b} + \frac{a}{( a + b ) ^{2}} + \frac{a}{( a + b ) ^{3}} + \dots = \frac{a / ( a + b )}{1 - 1/ ( a + b )} = \frac{a}{a + b - 1} = \frac{4 ( b - 1 )}{4 ( b - 1 ) + ( b - 1 )} = \frac{4 ( b - 1 )}{5 ( b - 1 )} = \frac{4}{5} .