NCERT Solutions
Class 11 Maths
Principle of Mathematical Induction
Ex.4.1 Q.17
Prove the following by using the principle of mathematical induction for all n є N:
+ 
+ 
+ …………. + [1 ÷ {(2n + 1) (2n + 3)}] = [n ÷ {3(2n + 3)}]
Let the given statement be P(n), i.e.,
P(n): 
+ 
+ 
+ …………. + 1 ÷ {(2n + 1) (2n + 3)} = n ÷ {3(2n + 3)}
For n = 1, we have
P (1): = 1 ÷ {3(2.1 + 3)} = , which is true.
Let P(k) be true for some positive integer k, i.e.,
P(k): + + + …………. + 1 ÷ {(2k + 1) (2k + 3)} = k ÷ {3(2k + 3)} ………… (1)
We shall now prove that P (k + 1) is true.
Consider
+ + + …………. + 1 ÷ {(2k + 1) (2k + 3)} + 1 ÷ [{2(k + 1) + 1} {(2(k + 1) + 3}]
= k ÷ {3(2k + 3)} + 1 ÷ [{2(k + 1) + 1} {(2(k + 1) + 3}]
[From equation 1]
= k ÷ {3(2k + 3)} + 1 ÷ {(2k + 3) (2k + 5)}
= {1 ÷ (2k + 3)} {
+ 
}
= {1 ÷ (2k + 3)} [{k (2k + 5) + 3} ÷ {3(2k + 5)}]
= {1 ÷ (2k + 3)} {2k2 + 5k + 3} ÷ {3(2k + 5)}
= {1 ÷ (2k + 3)} {2k2 + 2k + 3k + 3} ÷ {3(2k + 5)}
= {1 ÷ (2k + 3)} {2k (k + 1) + 3(k + 1)} ÷ {3(2k + 5)}
= {1 ÷ (2k + 3)} {(2k + 3) (k + 1)} ÷ {3(2k + 5)}
= (k + 1)} ÷ {3(2k + 5)}
= (k + 1)} ÷ [3{2(k + 1) + 3)}]
Thus, P (k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.