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Find the sum 2017+1/4(2016+1/4(2015+…+...

Find the sum
`2017+1/4(2016+1/4(2015+…+1/4(2+1/4(1))..))`

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Verified by Experts

We have
`S=2017+1/4xx2016+1/4^(2)xx2015+..+1/(4^(2016))xx1` …(1)
Clearly, this is A.G.P
`therefore1/4s=1/4xx2017+1/4^(2)xx2016+..+1/(4^(2016))xx2+1/(4^(2017))xx1` …(2)
Subracting (2) from (1), we get
`3/4S=2017-1/4-1/4^(2)-…-1/(4^(2016))-1/(4^(2017))`
`3/4S=2017-((1/4(1-1/4^(2017)))/((1-1/4)))`
`S=4/3(2017)-4/9(1-1/4^(2017))`
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