Menu Close

Given-the-equality-1-3-5-2p-1-p-1-2-p-N-Show-this-equality-is-true-when-we-replace-p-by-p-1-




Question Number 116685 by mathocean1 last updated on 05/Oct/20
Given the equality:  1+3+5+...+(2p+1)=(p+1)^(2 )    p ∈ N^∗     Show this equality is true when  we replace p by p+1
Giventheequality:1+3+5++(2p+1)=(p+1)2pNShowthisequalityistruewhenwereplacepbyp+1
Answered by TANMAY PANACEA last updated on 05/Oct/20
1+(n−1)2=2p+1  2n−1=2p+1→n=p+1  sum of (p+1) terms  ((p+1)/2)[2×1+(p+1−1)2]  (((p+1))/2)×[2p+2]=(p+1)^2   now sum of (p+1+1)terms from LHS  (((p+2))/2)[2×1+(p+2−1)×2]  =(((p+2))/2)×[2+2p+2]→(p+2)^2   RHS (p+1+1)^2 →(p+2)^2   hence proved
1+(n1)2=2p+12n1=2p+1n=p+1sumof(p+1)termsp+12[2×1+(p+11)2](p+1)2×[2p+2]=(p+1)2nowsumof(p+1+1)termsfromLHS(p+2)2[2×1+(p+21)×2]=(p+2)2×[2+2p+2](p+2)2RHS(p+1+1)2(p+2)2henceproved

Leave a Reply

Your email address will not be published. Required fields are marked *