+249 Consider two infinite geometric series. The first has leading term a, common ratio b, and sum S. The second has a leading term b, common ratio a, and sum S. Find the value of a+b.
+129852 Consider two infinite geometric series. The first has leading term a, common ratio b, and sum S. The second has a leading term b, common ratio a, and sum S. Find the value of a+b.
We have that
a / [ 1 - b] = S and b / [ 1 - a] = S
Therefore
a / [ 1 - b ] = b / [ 1 - a] cross-multiply
a [ 1 - a] = b [ 1 - b] simplify
a - a^2 = b - b^2 rearrange
( a - b ) = a^2 - b^2 factor the right side
(a - b) = (a - b) ( a + b) divide both sides by (a - b)
1 = a + b
