Copyright © University of Cambridge. All rights reserved.

'Mean Sequence' printed from https://nrich.maths.org/

Show menu


Answer: $\frac34$


Numbers
$\frac{2}{3}+\frac{4}{5}=\frac{10}{15}+\frac{12}{15}$, so the average is $\frac{11}{15}$

$\frac45+\frac{11}{15} = \frac{12}{15}+\frac{11}{15}=\frac{23}{15}$ so the average is $\frac{23}{30}$

$\frac{11}{15}+\frac{23}{30} = \frac{22}{30}+\frac{23}{30}=\frac{45}{30}$ so the average is $\frac{45}{60}=\frac34$



Algebra
Suppose the first two terms of the sequence are $x$ and $y$.

Third term is $\frac{1}{2}(x+y)$

Fourth term is $\frac12\left(y+\frac12(x+y)\right)=\frac{1}{4}(x+3y)$

Fifth term is $\frac12\left(\frac12(x+y)+\frac14(x+3y)\right)=\frac{1}{8}(3x+5y)$.

Putting $x=\frac{2}{3}$ and $y=\frac{4}{5}$ we obtain $\frac{1}{8}(2+4)=\frac{3}{4}$.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.