BITSAT Mathematics Question (2023) — Solution

Step-by-step solution

Let the statement P ( n ) be defined as P(n): n^5 5 + n^3 3 + 7 n 15 is a natural number for all n N : Step I : For n =1, P (1): 1 5 + 1 3 + 7 15 =1 N Hence, it is true for n =1. Step II : Let it is true for n=k, i.e. k^5 5 + k^3 3 + 7 k 15 = N (i) Step III : For n = k +1, ( k +1)^5 5 + ( k +1)^3 3 + 7( k +1) 15 = 1 5 (k^5+5 k^4+10 k^3+10 k^2+3 k+1 ) + 1 3 (k^3+3 k^2+3 k+1 )+ 7 15 k+ 7 15 = ( k^5 5 + k^3 3 + 7 15 k )+ (k^4+2 k^3+3 k^2+2 k ) + 1 5 + 1 3 + 7 15 = + k ^4+2 k ^3+3 k ^2+2 k +1 [using equation (i)] which is a natural number, since k N . Therefore, P(k+1) is true, when P(k) is true. Hence, from the principle of mathematical induction, the statement is true for all natural numbers n