Oxford University MAT 2016 Solved Papers

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Question : 7 of 10

Marks: +1, -0

The sequence

(x_n)

, where

n ≥ 0

, is defined by

x_0=1

and

x_n=∑↙{k=0}↖{n-1} x_k \;\; \;\text " for "\; n ≥ 1 .

The sum

∑↙{k=0}↖{∞} \;{1}/{x_k}

equals

1,
$\;{6}/{5}$ ,
$\;{8}/{5}$ ,
3,
$\;{27}/{5}$ .

Solution:

Considering the first few terms

x_0=1, x_1=x_0=1, x_2=2, x_3=4, x_4=8

, and so on. By observation,

x_n=2^{n-1}

for

n ≥slant 1

. As this is a geometric progression, we can evaluate the sum of the sequence as

∑↙{k=0}↖{∞} \;{1}/{x_k} \;\; =\;{1}/{1}+∑↙{k=1}↖{∞} \;{1}/{2^{k-1}}

\;\; =1+\;{1}/{1-\;{1}/{2}}

\;\; =3

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