Home
Class 12
MATHS
Prove that (25)^(n+1)-24n+5735 is divisi...

Prove that `(25)^(n+1)-24n+5735` is divisible by `(24)^2` for all `n=1,2, `

Text Solution

Verified by Experts

Let `P(n)=(25)^(n+1)-24n+5735`
Step I For `n=1`.
`P(1)=(25)^2-24+5735=625 -24+6336=11xx(24)^2`, which is divisible by `(24)^2`.
Therefore , the result is true for `n=1`.
Step II Assume that the result is true for `n=k`. Then , `P(k)=(25)^(k+1)-24k+5735` is divisible by `(24)^2`.
`rArr P(k)=(24)^2r`, where r is an integer .
Step III For `n=k+1`,
`P(k+1)=(25)^((k+1)+1) -24(k+1)+5735`
`=(25)^(k+2)-24k+5711`
`=(25)(25)^(k+1)-24k+5711`
Now , `P(k+1)-P(k)`
`={(25)(25)^(k+1)-24k+5711}-{(25)^(k+1)-24k+5735}`
`=(24)(25)^(k+1)-24`
`=24{(25)^(k+1)-1}`
`rArr P(k+1)=P(k)+24{(25)^(k+1)-1}`
But the assumption P(k) is divisible by `(24)^2`. Also ,`24{(25)^(k+1)-}` is clearly divisible by `(24)^2`, forall k in N`. This shows that , the result is true for `n=k+1`.
Hence , the principle of mathematical induction , result is true for all `n in N`.
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove that x^(2n-1)+y^(2n-1) is divisible by x+y

3^(2n)+24n-1 is divisible;

Prove that (n !) is divisible by (n !)^(n-1)!

If p is a fixed positive integer, prove by induction that p^(n +1) + (p + 1)^(2n - 1) is divisible by P^2+ p +1 for all n in N .

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

Prove by induction that the integer next greater than (3+sqrt(5))^n is divisible by 2^n for all n in N .

prove using mathematical induction, n(n+1)(n+5) is divisible by 6 for all natural numbers

Using binomial theorem , Prove that 2^(3n)-7n-1 is divisible by 49.

Prove the statement by the principle of mathematical induction : 2^(3n) - 1 is divisible by 7, 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.