Question
in the binomial expansion of (1+x/k)^n, where k is a constant and n is a positive integer, the coefficients of x and x^2 are equal.
(a) show that 2k=n-1
(b) deduce the value of k. Hence find the 1st 3 terms in the expansion in ascending powers of x.
(a) show that 2k=n-1
(b) deduce the value of k. Hence find the 1st 3 terms in the expansion in ascending powers of x.
Answers
(1+x/k)^n = 1^n + n(x/k) + n(n-1)/2 (x/k)^2
n/k = n(n-1)/2k^2
2kn/2k^2 = n(n-1)/2k^2
2kn = n(n-1)
2k = n-1
we know that n is odd, since n=2k+1
if k=1,
(1+x)^3 = 1+3x+3x^2+...
if k=2, we have
(1+x/2)^5 = 1+5(x/2) + 5*4/2*(x/2)^2+...
= 1 + 5/2 x + 5/2 x^2 + ...
For any integer value of k>0, n=2k+1
will work.
So, (b) is really a bogus question
n/k = n(n-1)/2k^2
2kn/2k^2 = n(n-1)/2k^2
2kn = n(n-1)
2k = n-1
we know that n is odd, since n=2k+1
if k=1,
(1+x)^3 = 1+3x+3x^2+...
if k=2, we have
(1+x/2)^5 = 1+5(x/2) + 5*4/2*(x/2)^2+...
= 1 + 5/2 x + 5/2 x^2 + ...
For any integer value of k>0, n=2k+1
will work.
So, (b) is really a bogus question
Related Questions
find the first 3 terms in the expansion of (x+4)(1+3x)^-2 as a series in ascending powers.
I expa...
Find
1.The first 4 terms of the binomial expansion in ascending powers of x of (1+x/4)^8.
2.Use y...
i) Write down the first three terms in the expansion of (2-x/2)^n in ascending powers of x, where n...
Write the four terms of the binomial expansion (2-1/x)^5 in ascending powers of X. Hence find (1.9)^...