+96 The real numbers a and b satisfy |a| < 1 and |b| < 1. (a) In a grid that extends infinitely, the first row contains the numbers 1, a, a^2, . . . . The second row contains the numbers b, ab, a^2 b, . . . . In general, each number is multiplied by a to give the number to the right of it, and each number is multiplied by b to give the number below it. Find the sum of all numbers in the grid.
(b) Now suppose the grid is colored like a chessboard, with alternating black and white squares, as shown below. Find the sum of all the numbers that lie on the black squares.
+23252 First problem:
The first row is a geometric series with first term 1 and common ratio a whose sum is 1/(1 - a).
The second row is also geometric with first term b and common ratio a whose sum is b/(1 - a).
The third row is also geometric with first term b2 and common ratio a whose sum is b2/(1 - a).
The sum of the rows is: 1/(1 - a) + b/(1 - a) + b2/(1 - a) + ... = 1/(1 - a) · [ 1 + b + b2 + ... ]
The series 1 + b + b2 + ... is geometric with first term 1 and common ratio b whose sum is 1/(1 - b).
Therefore, the sum of the rows is: 1/(1 - a) · [ 1 + b + b2 + ... ] = 1/(1 - a) · 1/(1 - b) = 1 / [ (1 - a)(1 - b) ]
+23252 Second problem:
1 + a2 + a4 + ... = 1/(1 - a2)
ab + a3b + a5b + ... = ab/(1 - a2)
b2 + a2b2 + a4b2 + ... = b2/(1 - a2)
ab3 + a3b3 + a5b3 + ... = ab3/(1 - a2)
...
Sum = 1/(1 - a2) + ab/(1 - a2) + b2/(1 - a2) + ab3/(1 - a2) + ...
= 1/(1 - a2) · [ 1 + ab + b2 + ab3 + b4 + ab5 + b6 + ab7 + ...]
= 1/(1 - a2) [ (1 + b2 + b4 + b6 + ...) + (ab + ab3 + ab5 + ab7 + ...) ]
= 1/(1 - a2) · [ ( 1/(1 - b2 ) + ( ab/(1 - b2) ) ]
= [ 1/(1 - a2) · 1/(1 - b2 ) ] · [ 1 + ab ]
= ( 1 + ab ) / [ (1- a2)(1 - b2) ]
