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Proof by induction help please!? ((1-(1/2^2)) * (1-(1/3^2)) * 1-(1/4^2)) ......
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High School Teacher
From the given, we notice that the smallest value n can take is 2.
So when n=2, we have `1-1/(2^2)=1-1/4=3/4` and `(2+1)/[2*2]=3/4`
Hence the statement is true for n=2. Now suppose that the statement is true for n, we need to prove it for n+1. In other words we need to prove the following equality:
Let's simplify the right hand side = `(n+2)/[2(n+1)]`
Since we are assuming that the statement is true for n, then we know that `(1-1/[2^2])*(1-1/[3^2])*...*(1-1/[n^2])=(n+1)/[2n]`
If we substitue the following in the Left Hand side of the equation we are proving we get
`[(n+1)*n*(n+2)]/[2n(n+1)^2]=` If we simplify, we get
`(n+2)/[2(n+1)]` Which is equal to the right hand side.
Hence the statement is proved by induction.
Posted by rcmath on May 14, 2012 at 5:17 PM (Answer #1)
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