Advertisements
Advertisements
Question
If a and b are distinct integers, prove that a − b is a factor of an − bn, whenever n is a positive integer. [Hint: write an = (a − b + b)n and expaand]
Advertisements
Solution
a = a − b + b
So, an = [a − b +b]n
= [(a − b) + b]n
= nC0 (a − b)n + nC1 (a − b)n−1b1 + nC2 (a − b)n−2b2 + ....... + nCn−1 (a − b)bn−1 + nCn (bn)
⇒ an − bn = (a − b)n + nC1 (a − b)n−1b + nC2 (a − b)n−2b2 + ....... + nCn−1 (a − b)bn−1
= (a − b) [(a − b)n−1 + nC1 (a − b)n−2b + nC2 (a − b)n−3b2 + ...... + nCn−1 bn−1]
= (a – b)[an integer]
⇒ an – bn is divisible by (a – b)
APPEARS IN
RELATED QUESTIONS
Evaluate the following using binomial theorem:
(101)4
Evaluate the following using binomial theorem:
(999)5
Expand the following by using binomial theorem.
`(x + 1/y)^7`
Find the 5th term in the expansion of (x – 2y)13.
Find the term independent of x in the expansion of
`(x^2 - 2/(3x))^9`
Find the term independent of x in the expansion of
`(x - 2/x^2)^15`
Find the term independent of x in the expansion of
`(2x^2 + 1/x)^12`
Show that the middle term in the expansion of is (1 + x)2n is `(1*3*5...(2n - 1)2^nx^n)/(n!)`
Find the Co-efficient of x11 in the expansion of `(x + 2/x^2)^17`
The constant term in the expansion of `(x + 2/x)^6` is
The last term in the expansion of (3 + √2 )8 is:
Compute 1024
If n is a positive integer, using Binomial theorem, show that, 9n+1 − 8n − 9 is always divisible by 64
If n is a positive integer and r is a non-negative integer, prove that the coefficients of xr and xn−r in the expansion of (1 + x)n are equal
In the binomial expansion of (a + b)n, if the coefficients of the 4th and 13th terms are equal then, find n
If the binomial coefficients of three consecutive terms in the expansion of (a + x)n are in the ratio 1 : 7 : 42, then find n
In the binomial expansion of (1 + x)n, the coefficients of the 5th, 6th and 7th terms are in AP. Find all values of n
Prove that `"C"_0^2 + "C"_1^2 + "C"_2^2 + ... + "C"_"n"^2 = (2"n"!)/("n"!)^2`
