Logarithm and Infinite sequence

Algebra Level 2

Solve for Y Y .

log ⁡ 4 Y = 1 + 1 3 + 1 9 + . . . \log _{ 4 }{ Y } =1+\frac { 1 }{ 3 } +\frac { 1 }{ 9 } +...


The answer is 8.

Horace Pang by Horace Pang , 20, Hong Kong

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1 solution

Joe Mansley
Jul 19, 2015

We have ourselves an infinite geometric series with a 1st term of 1 and a common ratio of 1/3. The sum of an infinite geometric series is a/(1-r)=1/(1-1/3)=3/2. So we have log4(Y)=3/2. Y=4^(3/2)=8

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Just one thing, we have Y= (4)^3/2 where Y=8 or -8 if we only take into account the context of the above equation. But since we also have log 4 (Y) where we know for any log 4 (x), x>0, therefore we must reject the negative answer hence only answer left is +8

Jeffrey Li - 5 years, 10 months ago

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