Question

The first three terms in the expansion of (2+ax)^n in ascending powers of x are 32-40x+bx^2. Find the values of the constants n, a and b

Answer 1

the value of n is 5

since 2^5 = 32

so the expansion is

\[(2 + ax)^n =^nC_{r} (2)^{n - r}(ax)^r\]

using the expansion with n = 5 and r = 0 you get

\[^5C_{0}(2)^5(ax)^0 = 32\]

so to find a, find the coefficient of the 2nd term using the expansion where r = 1

\[^5C_{1}(2)^{5 -1}(ax)^1 = 5 \times 2^4 \times ax\]

then you have

\[5 \times 16 \times ax = 40ax\]

now equate the coefficients

-40 = 40a

therefore a = -1

so your binomial is now

\[(2 - x)^5\]

to find the value of b, let r = 2

then

\[^5C_{2} (2)^{5-2} ( -x)^2\]

just evaluate then compare

hope this helps

Answer 2

amazing, thanks so much!

Answer 3

lol..glad to help