A converging sequence

Calculus Level 1

The sequence $\dfrac12, \dfrac14, \dfrac38, \dfrac5{16} , \dfrac{11}{32} , \dfrac{21}{64} , \ldots$ converges on which number?

7/20 1/3 9/28 3/10

2 solutions

Kattos Kattos
Feb 21, 2016

The sequence is $a_n/b_n$ where $a_n+a_{n+1}=b_n$ and $b_n=2 b_{n-1}$ . Divide the first equation by $b_{n+1}=2 b_n$ and take the limit. You get $x/2 +x=1/2$ . Therefore the limit is $x=1/3$ .

Denton Young
Feb 8, 2016

The sequence alternately adds and subtracts $1/(2^n)$ from the terms. The limit is therefore 1/4 + 1/16 + 1/64... = 1/4 (4/3) = 1/3

Moderator note:

The pattern in the sequence is not immediately obvious to me. Is there a way to make this clearer?

Calvin Lin Staff - 5 years, 3 months ago

Try focusing on the fact that the denominators are consecutive powers of 2, and the numerators are alternately 2n+1 and 2n-1, where n was the previous numerator...

Denton Young - 5 years, 3 months ago

Can you add that information to the problem? Thanks!

Calvin Lin Staff - 5 years, 3 months ago

Geometric Progression

S1 = a = 1/2

S2 - S1 = 1/4 - 1/2 = -1/4

S3 - S2 = 3/8 - 1/4 = 1/8

S4 - S3 = 5/16 - 3/8 = -1/16

Therefore, r = -1/2

S = a/(1-r) = 0.5/1.5 = 1/3

Albert Thaw Tun - 5 years, 3 months ago

HOW DO YOU USE SUM OF GP FORMULA IF THE DIFFERENCES ARE IN G.P.?PLEASE EXPLAIN ME.

Dhiraj Kushwaha - 5 years, 3 months ago

A converging sequence

2 solutions

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