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Jonask
A sequence of numbers (a_n) is defined as
\[a_1=1/2\] and for each \[n \ge 2\] \[\huge a_n=\left(\frac{ 2n-3 }{ 2n }\right)a_{n-1}\] Prove that \[\huge\sum_{k=1}^{n}a_k<1 \] for all \[n \ge1\]
|dw:1357738064697:dw|Just try to Prove
\[a_{1}+ a_{2}+…+ a_{n} < 1\] \[\frac{ 1 }{ 2 } + a_{2} + a_{3}+…+(\frac{ 2n - 3 }{ 2n })a_{n-1} < 1\] For n=1 LHS=\[\frac{ 1 }{ 2 }\] \[<RHS\] (Therefore) True for n=1 For n=2 LHS=\[(\frac{ 2(2) - 3 }{ 2(2) })a_{2-1}\] \[<RHS\] (Therefore) True for n=2
Assume the formula is true for n=m \[\frac{ 1 }{ 2 } + a_{2} + a_{3}+…+(\frac{ 2m - 3 }{ 2m })a_{m-1} < 1\] For n=m+1 \[\frac{ 1 }{ 2 } + a_{2} + a_{3}+…+(\frac{ 2m - 3 }{ 2m })a_{m-1}+(\frac{ 2(m+1) - 3 }{ 2(m+1) })a_{m+1-1} < 1\] LHS= \[1 + (\frac{ 2(m+1) - 3 }{ 2(m+1) })a_{m}\] \[=(\frac{ 2m-1 }{ 2m+2 })(\frac{ 2m - 3 }{ 2m })a_{m-1}\]
@sauravshakya @azteck thanks
I am not sure if this helps
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Did u mean like this @experimentX
yeah .. somewhat similar to that. but one terms is missing.
there should be one n ... or i made mistake.
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It was me who did a mistake
hmm ... now lets use this identity. http://planetmath.org/GeneratingFunctionForTheCatalanNumbers.html