the problem given is TT i=0 to n (1/(2i+1)*(1/2i+2) = 1(2n+2)!

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Hello im working on a proof by induction and I need something clearified. 

the problem given is TT i=0 to n (1/(2i+1)*(1/2i+2) = 1(2n+2)!

my question is how does 1/(2K+2)!(2k+3)(2k+4) become 1/(2k+4)! 

If possible, please break down step by step so that I can undertand. I really want to know how to to solve this problem. 

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