Summing Up

Algebra Level 1

Evaluate

1 + 1 2 + 1 4 + 1 8 + 1 16 + … \large 1 + \dfrac{1}{2}+\dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \dots


The answer is 2.

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Florencio Caballero by Florencio Caballero , 45, Philippines

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4 solutions

Akshat Sharda
Oct 7, 2015

x = 1 + 1 2 + 1 2 2 + 1 2 3 + 1 2 4 + … ∞ x = 1 + 1 2 ( 1 + 1 2 + 1 2 2 + … ∞ ) x = 1 + 1 2 x ⇒ 2 x = 2 + x ⇒ x = 2 x=1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\frac{1}{2^{4}}+\ldots \infty \\ x=1+\frac{1}{2} \left(1+\frac{1}{2}+\frac{1}{2^{2}}+\ldots \infty \right) \\ x=1+\frac{1}{2}x \Rightarrow 2x=2+x \Rightarrow \boxed{x=2}

13 Helpful 0 Interesting 0 Brilliant 0 Confused

Excellent!

Swapnil Das - 5 years, 8 months ago

Let n = 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . So 2n = 2 + 1 + 1/2 + 1/4 + 1/8 + . . . Therefore 2n = 2 + n. It follows that n = 2.

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Muhammad Maulana
Oct 9, 2015

S~= a/1-r = 1/(1-0.5) = 2

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Alok Jindal
Nov 8, 2015

Its a case of infinite G.P. so simply apply formula a/(1-r) Where a = 1 & r= 1/2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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