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C2 exam question please help me ergent thanks

I dunno what to do i would be gratefull if someone was to help me. Thanks

The first four terms, in ascending powers of x, of the binomial expansion of (1 + kx)n are
1 + Ax + Bx2 + Bx3 + …,
where k is a positive constant and A, B and n are positive integers.

(a) By considering the coefficients of x2 and x3, show that 3 = (n – 2) k.

Given that A = 4,
(b) find the value of n and the value of k.

if anyone has got a marksheme for mock paper edexcel C2
it would be great

Reply 1

danniella

Hello!

(1+kx)^n = 1^n + nC1(kx) + nC2 (kx)^2 + nC3 (kx)^3

where nCr = n!/((n-r!)r!)
nC1 = n
nC2 = n(n-1)/2
nC3 = n(n-1)(n-2)/(3x2) = n(n-1)(n-2)/6

so the first four terms of your expansion are
1 + nkx + n(n-1)/2 k^2x^2 + n(n-1)(n-2)/6 k^3x^3

The coefficients of the third and fourth terms are both called B in the question, so they are equal.
n(n-1)/2 k^2 = n(n-1)(n-2)/6 k^3
Multiply the whole thing by 6
3n(n-1)k^2 = n(n-1)(n-2)k^3

Cancel the n, the n-1 and the k^2 (as n=0, k=0, n=1 are not possible in the context of the question)
so
3=(n-2)k

Rest of question to follow.

love danniella

love danniella

Reply 2

danniella

Hello!

Second part.

The coefficient of the second term is nk = A (given in the question)
so nk = 4
But you've just worked out that
3=k(n-2)
3=nk-2k

But nk=4
so 3=4-2k
So 2k=1 and k=0.5

If nk=4, then 4=0.5n and hence n=8

So k=0.5 and n=8

love danniella

Reply 3

Rajeshshaym