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'Mean Sequence' printed from https://nrich.maths.org/
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 $\frac{1}{8}(2+4)=\frac{3}{4}$.