Question

Find the generating function and the general term of the following series: $3+6x+14x^2+36x^3+98x^4+276x^5+....$

Answer 1

$3+6x+14x^2+36x^3+98x^4+276x^5+\cdot \cdot \cdot$

$3=(1^0+2^0+3^0)x^0$

$6x=(1^1+2^1+3^1)x^1$

$14x^2=(1^2+2^2+3^2)x^2$

$98x^3=(1^3+2^3+3^3)x^3$

Hence, the general term is $(1^{n+1}+2^{n+1}+3^{n+1})x^{n+1}$