To Prove:
\(\frac{1}{1\times3}\) + \(\frac{1}{(3\times5)}\) +......+ \(\frac{1}{(2n-1)\times(2n+1)}\) = \(\frac{n}{(2n+1)}\)
Let us prove this question by principle of mathematical induction (PMI)
Let P(n):\(\frac{1}{1\times3}\) + \(\frac{1}{(3\times5)}\) +......+ \(\frac{1}{(2n-1)\times(2n+1)}\) = \(\frac{n}{(2n+1)}\)
For n = 1
LHS = \(\frac{1}{1\times3}\) = \(\frac{1}{3}\)
RHS = \(\frac{1}{(2+1)}\) = \(\frac{1}{3}\)
Hence, LHS = RHS
P(n) is true for n = 1
Assume P(k) is true
\(\frac{1}{1\times3}\) + \(\frac{1}{(3\times5)}\) +......+ \(\frac{1}{(2k-1)\times(2k+1)}\) = \(\frac{k}{(2k+1)}\).....(1)
We will prove that P(k + 1) is true
[Writing the second last term]
(Splitting the numerator and cancelling the common factor)
= RHS
LHS = RHS
Therefore, P (k + 1) is true whenever P(k) is true
By the principle of mathematical induction, P(n) is true for×
Where n is a natural number
Hence proved.