Question

Prove using mathematical induction that 3" + 4" < 5" for all n > 2.

Answer 1

Solution:

To prove: 3^n + 4^n $\leq$ 5^n for all n $\geq$ 2

Explanation:

Proving using Mathematical Induction:

Step 1:

=>Checking for base condition n = 2

Put n = 2

=>LHS = 3^2 + 4^2

=>LHS = 9 + 16

=>LHS = 25

=>RHS = 5^2

=>RHS = 25

=>Hence LHS = RHS for base condition so 3^n + 4^n $\leq$ 5^n is true for base condition.

Step 2:

=>Assuming the expression 3^n + 4^n $\leq$ 5^n is true for n = k

=>3^k + 4^k $\leq$ 5^k....(1)

Step 3:

=>Checking the expression for n = k+1

=>LHS = 3^(k+1) + 4^(k+1)

=>LHS = 3*3^k + 4*4^k

=>LHS = (3^k + 4^k) + 2*3^k + 3*4^k

=>Taking equal(=) sign from equation (1) so we can write 3^k + 4^k = 5^k

=>LHS = 5^k + 2*3^k + 3*4^k

=>RHS = 5^(k+1)

=>RHS = 5*5^k

=>RHS = 5^k + 4*5^k

=>We know that expression 4*5^k > 2*3^k + 3*4^k

=>Hence we can write 5^k + 2*3^k + 3*4^k $\leq$ 5^k + 4*5^k

=>Hence the expression is true for n = k+1

=>Hence on the basis of above statements we have proved our expression using Mathematical Induction.

I have explained each and every part with the help of statements attached to it.