I will post a challenging math problem each day. The level of difficulty will vary, but most problems should not require any specialized knowledge beyond calculus. Problems are also posted on Twitter using the hashtag #mathpotd.
Sunday, September 27, 2009
A Fibonacci sum
Evaluate the sum of Fib(n)/10^n where Fib(n) denotes the nth Fibonacci number.
That is, calculate 1/10 + 1/10^2 + 2/10^3 + 3/10^4 + 5/10^5 + 8/10^6 + 13/10^7 + ...
S = sum (1 to inf) Fib(n)/10^n = 1/10 + sum (2 to inf) Fib(n)/10^n = 1/10 + sum (2 to inf) (Fib(n-1)+Fib(n-2))/10^n = 1/10 + sum (2 to inf) Fib(n-1)/10^n + sum (2 to inf) Fib(n-2)/10^n = 1/10 + sum (1 to inf) Fib(n)/10^(n+1) + sum (0 to inf) Fib(n)/10^(n+2) = 1/10 + S/10 + S/100
Answer: 10/89
ReplyDeleteS = sum (1 to inf) Fib(n)/10^n
= 1/10 + sum (2 to inf) Fib(n)/10^n
= 1/10 + sum (2 to inf) (Fib(n-1)+Fib(n-2))/10^n
= 1/10 + sum (2 to inf) Fib(n-1)/10^n + sum (2 to inf) Fib(n-2)/10^n
= 1/10 + sum (1 to inf) Fib(n)/10^(n+1) + sum (0 to inf) Fib(n)/10^(n+2)
= 1/10 + S/10 + S/100
Solving S = 1/10 + S/10 + S/100 gives S = 10/89.