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jmc

algebra senior

Problem

Let

a_{1}, a_{2}, \dots

be a sequence defined by

a_{1} = a_{2} = 1

and

a_{n + 2} = a_{n + 1} + a_{n}

for

n \geq 1

. Find

n = 1 \sum \infty \frac{a _{n}}{4 ^{n + 1}} .

Solution

Let

X

denote the desired sum. Note that

X 4 X 16 X = \frac{0}{4 ^{0}} + \frac{0}{4 ^{1}} + \frac{1}{4 ^{2}} + \frac{1}{4 ^{3}} + \frac{2}{4 ^{4}} + \frac{3}{4 ^{5}} + \frac{5}{4 ^{6}} + \dots = \frac{0}{4 ^{0}} + \frac{1}{4 ^{1}} + \frac{1}{4 ^{2}} + \frac{2}{4 ^{3}} + \frac{3}{4 ^{4}} + \frac{5}{4 ^{5}} + \frac{8}{4 ^{6}} + \dots = \frac{1}{4 ^{0}} + \frac{1}{4 ^{1}} + \frac{2}{4 ^{2}} + \frac{3}{4 ^{3}} + \frac{5}{4 ^{4}} + \frac{8}{4 ^{5}} + \frac{13}{4 ^{6}} + \dots

so that

X + 4 X = 16 X - 1

, and

X = \frac{1}{11}

.

Final answer

\frac{1}{11}