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