- If (1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20) then find the coefficients of a_1 ; a_2 ; a_3 and a_20
Solution
From our hypothesis we already had:
(1+2x+3x^2)^10=a_0+a_1x+a_2x^2+.......+a_(20)x^(20)
We will find the coefficients of a_1 ; a_2 ; a_3 and a_20
Which meant they are the coefficients of x ; x^2 ; x^3 and x^(20) .
We can rewrite (1+2x+3x^2)^10=[1+x(2+3x)]^10
We will use the Newton's Formula
(a+b)^n=\sum_{i=0}^nC(n,i)a^(n-i).b^i
Then, [1+x(2+3x)]^10=C(10,0)+C(10,1)x(2+3x)+C(10,2)x^2.(2+3x)^2
+C(10,3)x^3.(2+3x)^3+....+C(10,10)x^10.(2+3x)^10
We observe that:
Coef(x)=2C(10,1)=20 then, a_1=20
Coef(x^2)=2^2C(10,2)=4.45=180 then, a_2=180
Coef(x^3)=12C(10,2)+8C(10,3)=45.12+120.8=1500 then a_3=1500
Coef(x^20)=C(10,10).3^10=3^10 then, a_20=3^10