by PMI prove , 1/1 3 + 1/3 5+ 1/5 7+ - Maths - Principle of Mathematical Induction

Question

by PMI prove , 1/1 3 + 1/3 5+ 1/5 7+ 1/(2n-1)(2n+1) =
n/(2n+1)

Niharika Khare · Accepted Answer

Let a statement P(n) be given by:P(n) : 11
3+13
5+-----+12n-12n+1=n2n+1Step1: To prove that the statement is true for n=1 i
e  P(1) is true
              LHS=11
3=13             RHS=12×1+1=13Thus, P(1) is true
Step2 : Assume that the statement is true for n=k, where k is some positive integer
             11
3+13
5+-----+12k-12k+1=k2k+1Step3: Prove that the statement is true for n=k+1             LHS= 11
3+13
5+-----+12k-12k+1+12k+1-12k+1+1                  =k2k+1+12k+1-12k+1+1                  =k2k+1+12k+12k+3=2k2+3k+12k+12k+3=2k+1k+12k+12k+3=k+12k+3             RHS=k+12k+1+1=k+12k+3⇒LHS=RHS for n=k+1⇒P(k+1) is true whenever P(k) is true
 Thus from the principle of mathematical induction P(n) is true for all natural numbers n

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