Home
Class 12
MATHS
By mathematical induction prove that 2^(...

By mathematical induction prove that `2^(3n)`-1 is divisible by 7.

Text Solution

Verified by Experts

`2^(3n) - 1 = (2^(3))^(n) - 1 = (1+7)^(n) - 1`
`= [1+.^(n)C_(1)(7) + .^(n)C_(2)(7)^(2) + "……" + .^(n)C_(n) (7)^(n)] - 1`
`= 7 [.^(n)C_(1) + .^(n)C_(2) (7) + "….." + .^(n)C_(n) (7)^(n-1)]`
Thus, `2^(3n) - 1` is divisible by 7 for all `n in N`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.3|7 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.4|13 Videos

  • BINOMIAL THEOREM

    CENGAGE|Exercise Exercise 8.1|17 Videos

  • AREA UNDER CURVES

    CENGAGE|Exercise Question Bank|10 Videos

  • CIRCLE

    CENGAGE|Exercise MATRIX MATCH TYPE|6 Videos

Similar Questions

Explore conceptually related problems

Using principle of mathematical induction, prove that 7^(4^(n)) -1 is divisible by 2^(2n+3) for any natural number n.

Prove that 3^(2n)-1 is divisible by 8.

By mathematical induction show that 7^(2n)+16n-1 is divisible by 64.

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

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

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

Using binomial theorem, prove that 2^(3n)-7^n-1 is divisible by 49 , where n in Ndot

Use induction to prove that n^(3) - n + 3 , is divisible by 3, for all natural numbers n