Beyond infinity and more

Calculus Level 5

If sum of the infinite series $\dfrac12 + \dfrac28 + \dfrac{3}{16} + \dfrac{5}{32} + \dfrac{8}{64}+ \dfrac{13}{128} + \cdots = \dfrac{A}{B},$

where $\gcd(A,B)=1$ , then find $A+B$ .

Assumptions and Clarifications

$\dfrac{1}{4}$ between $\dfrac12$ and $\dfrac28$ is deliberately omitted.
With the exception of the missing $\frac{1}{4}$ term, the numbers in the numerator for the Fibonacci sequence and the numbers in the denominator are terms of a G.P.

1 solution

Milind Prabhu
Apr 8, 2016

Let $\large S=\sum _{ k=1 }^{ \infty }{ \frac { { F }_{ k } }{ { 2 }^{ k } } }$ where ${ F }_{ k }$ is the $k$ th term of the fibonacci sequence.

$\large \dfrac S2=\sum _{ k=1 }^{ \infty }{ \frac { { F }_{ k } }{ { 2 }^{ k+1 } } }$

$\large S-\dfrac S2=\dfrac 12+\dfrac S4$

Therefore $\large S=2$ . The sum we are interested in has the value $\large S-\dfrac 14=\dfrac 74$ . The answer is $\large 7+4=\boxed { 11 }$

Did the same way! Also try this .

Shanthanu Rai - 5 years, 2 months ago