Question

by PMI prove , 1/1.3 + 1/3.5+ 1/5.7+.......... 1/(2n-1)(2n+1) = n/(2n+1)

Answer 1

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Answer 2

Answer:

Hence it is proved by PMI that both sides are equal.

Step-by-step explanation:

Step 1: Given data  

P(n) : 1/1.3+1/3.5+−−−−−+1/(2n−1)(2n+1)=n/(2n+1)

Step 2: To prove, P(1) is true.              

LHS=1/1.3=1/3              

RHS=1/2×1+1=1/3

Thus, P(1) is true.

Step 3: Assume that the statement is true for n = k, where k is some positive integer.          

  1/1.3+1/3.5+−−−−−+1/(2k−1)(2k+1)=k/(2k+1)

Step 4:   for n=k+1            

 LHS= 1/1.3+1/3.5+−−−−−+1/(2k−1)(2k+1)+1/(2(k+1)−1)(2(k+1)+1)                  

=k/(2k+1)+1/(2(k+1)−1)(2(k+1)+1)              

 =k/(2k+1)+1/(2k+1)(2k+3)

=2k2+3k+1/(2k+1)(2k+3)

=(2k+1)(k+1)(2k+1)/(2k+3)

LHS=(k+1)/(2k+3)              

Step 5:

RHS=k+1/2(k+1) + 1)

=k+1 / 2k + 3

⇒LHS=RHS