Home
Class 12
MATHS
Using the principle of mathematical indu...

Using the principle of mathematical induction to show that `41^n-14^n` is divisible by 27 for all n.

Text Solution

Verified by Experts

Let`P(n):(1)/(1+x)+(2)/(1+x^2)+(4)/(1+x^4)+.....+(2^n)/(1+x^(2^(n)))`
`=(1)/(x-1)+(2^(n+1))/(1-x^(2^(n+1)))` .....(i)
Step I For `n=1`,
LHS of Eq. (i) =(1)/(1+x)+(2)/(1+x^2)`
Promotional Banner

Topper's Solved these Questions

  • MATHEMATICAL INDUCTION

    ARIHANT MATHS|Exercise Mathematical Induction Exercise 1: (Single Option Correct Tpye Questions)|3 Videos

  • MATHEMATICAL INDUCTION

    ARIHANT MATHS|Exercise Exercise (Statement I And Ii Type Questions)|3 Videos

  • LOGARITHM AND THEIR PROPERTIES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|4 Videos

  • MATRICES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|49 Videos

Similar Questions

Explore conceptually related problems

Use the principle of mathematical induction to show that a^(n) - b^n) is divisble by a-b for all natural numbers n.

Use the principle of mathematical induction to show that 5^(2n+1)+3^(n+2).2^(n-1) divisible by 19 for all natural numbers n.

Use the Principle of Mathematical Induction to prove that n(n + 1) (2n + 1) is divisible by 6 for all n in N.

Prove the following by using the principle of mathematical induction for all n in N :- x^(2n)-y^(2n) is divisible by x + y .

Prove the following by the principle of mathematical induction: \ 11^(n+2)+12^(2n+1) is divisible 133 for all n in Ndot

Prove the following by using the principle of mathematical induction for all n in N :- 10^(2n-1) + 1 is divisible by 11.

Prove the following by the principle of mathematical induction: \ x^(2n-1)+y^(2n-1) is divisible by x+y

Prove the following by using the principle of mathematical induction for all n in N :- 41^n- 14^n is a multiple of 27.

Prove the following by using the principle of mathematical induction for all n in N :- 3^(2n+2)-8n -9 is divisible by 8.