Answer:
n = 6, a = 5, b = 60
Step-by-step explanation:
In a binomial function (a + b)ⁿ expression that represents the terms,
(a + b)ⁿ = [tex]\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k[/tex]
By this formula,
1st term = [tex]\binom{n}{0}a^{n-0}b^0[/tex] = aⁿ
2nd term = [tex]\binom{n}{1}a^{n-1}b^1[/tex] = [tex]n.a^{n-1}b^1[/tex]
3rd term = [tex]\binom{n}{2}a^{n-2}b^2[/tex] = [tex]\frac{n(n-1)}{2}.a^{n-2}.b^2[/tex]
For the binomial expansion initial 3 terms of (2 - ax)ⁿ = 64 - 16bx + 100bx²
Terms of (2 - ax)ⁿ = [tex]2^n+n(2)^{n-1}(-ax)+\frac{n(n-1)}{2}(2)^{n-2}(-ax)^2[/tex]
= [tex]2^n-n(2)^{n-1}(ax)+n(n-1)(2)^{n-3}(a^2x^2)[/tex]
Comparing the terms of both the expansions,
1st term
2ⁿ = 64
2ⁿ = 2⁶
n = 6
2d term
[tex]n(2)^{n-1}(ax)=16bx[/tex]
[tex]6(2)^{6-1}(a)=16b[/tex]
192a = 16b
b = 12a -----(1)
3rd term
[tex]n(n-1)2^{n-3}(a^2x^2)=100bx^2[/tex]
[tex]6(6-1)2^{(6-3)}(a^2)=100b[/tex]
[tex]30(2)^3(a^2)=100b[/tex]
240a² = 100b
b = 2.4a² -----(2)
From equation (1) and (2),
b = 12a = 2.4a²
a = [tex]\frac{12}{2.4}=5[/tex]
From equation (1)
b = 12a = 60
Therefore, n = 6, a = 5, b = 60
