Binomial Theorem
Coefficient identity; roots of unity filter
MJMT_Full_Test_03
Grade 12
Question:
If $(1+x+x^2)^n=\sum_{r=0}^{2n}a_r x^r$, then $a_r-{}^nC_1 a_{r-1}+{}^nC_2 a_{r-2}-\cdots+(-1)^r{}^nC_r$ equals (r not multiple of 3)
Step-by-Step Solution
Key Concept: The sum is $[x^r]$ in $(1+x+x^2)^n(1-x)^n=((1-x^3))^n=\sum(-1)^k{}^nC_k x^{3k}$. Coefficient of $x^r$ is $0$ when $r\not\equiv0\pmod3$.
0.
Correct Answer: 1