Let [x] denote the greatest integer x.

Question

Let $[x]$ denote the greatest integer $\le x$. If for $n\in N$, $(1-x+x^3)^n=\displaystyle\sum_{j=0}^{3n}a_jx^j$, then $\displaystyle\sum_{j=0}^{\lfloor3n/2\rfloor}a_{2j}+4\sum_{j=0}^{\lfloor(3n-1)/2\rfloor}a_{2j+1}$ is equal to

Accepted Answer

Putting $x=1$: $(1-1+1)^n=1=a_0+a_1+\cdots+a_{3n}$ ...(i)

Putting $x=-1$: $(1+1-1)^n=1=a_0-a_1+a_2-\cdots$ ...(ii)

Adding: $2=2(a_0+a_2+a_4+\cdots)\Rightarrow a_0+a_2+a_4+\cdots=1$

Subtracting: $0=2(a_1+a_3+\cdots)\Rightarrow a_1+a_3+\cdots=0$

$$\sum a_{2j}+4\sum a_{2j+1} = 1+4(0)=1$$

Solutions · 1