If 1+x^4+x^5=_i=0^5a_i(1+x)^i, for all x in R, then a_2 is:

Question

If $1+x^4+x^5=\displaystyle\sum_{i=0}^5a_i(1+x)^i$, for all $x$ in $\mathbb R$, then $a_2$ is:

Accepted Answer

Expanding $\displaystyle\sum_{i=0}^5a_i(1+x)^i$ in powers of $x$ and comparing coefficients with $1+x^4+x^5$ gives the system:

$$a_5=1,\quad a_4+5a_5=1,\quad a_3+4a_4+10a_5=0,\quad a_2+3a_3+6a_4+10a_5=0$$

From these: $a_5=1\Rightarrow a_4=-4\Rightarrow a_3=6\Rightarrow a_2=-4$

Solutions · 1