Question

If n ∈ N and $(1 + x + x^2)^n = \sum_{r=0}^{2n} a_r x^r$, then $\sum_{r=0}^{n} (-1)^r a_r {}^nC_r$ is equal to

• A. 0 if n = 57
• B. 0 if n = 77
• C. ${}^{24}C_8$ if n = 24
• D. ${}^{39}C_{13}$ if n = 39

Answer 1

Answer: (B), (C), (D)

(B), (C), (D)

Answer 2

Let the given equation be $(1 + x + x^2)^n = \sum_{r=0}^{2n} a_r x^r$. We are asked to find the value of the sum $S = \sum_{r=0}^{n} (-1)^r a_r {}^nC_r$.

Consider the product of the series expansion of $(1+x+x^2)^n$ and the series expansion of $(1-1/x)^n$. $(1+x+x^2)^n = \sum_{r=0}^{2n} a_r x^r$. $(1-1/x)^n = \sum_{k=0}^n {}^nC_k (-1/x)^k = \sum_{k=0}^n (-1)^k {}^nC_k x^{-k}$.

The product is $(1+x+x^2)^n (1-1/x)^n = \left(\sum_{r=0}^{2n} a_r x^r\right) \left(\sum_{k=0}^n (-1)^k {}^nC_k x^{-k}\right)$. The coefficient of $x^0$ (the constant term) in this product is obtained by summing the products of terms $a_r x^r$ and $(-1)^k {}^nC_k x^{-k}$ such that $r-k=0$, i.e., $r=k$. The coefficient of $x^0$ is $\sum_{r=0}^{\min(2n, n)} a_r (-1)^r {}^nC_r$. Since the upper limit of the sum we want to evaluate is $n$, and ${}^nC_r=0$ for $r>n$, we can write the constant term as $\sum_{r=0}^{2n} a_r (-1)^r {}^nC_r$. As ${}^nC_r=0$ for $r>n$, this sum is equal to $\sum_{r=0}^{n} a_r (-1)^r {}^nC_r = \sum_{r=0}^{n} (-1)^r a_r {}^nC_r = S$.

Now let's evaluate the product $(1+x+x^2)^n (1-1/x)^n$ in a different way. $(1+x+x^2)^n (1-1/x)^n = (1+x+x^2)^n \left(\frac{x-1}{x}\right)^n = \frac{(1+x+x^2)^n (x-1)^n}{x^n}$. Using the identity $(x-1)(x^2+x+1) = x^3-1$, we have $(x-1)(1+x+x^2) = x^3-1$. So, the expression becomes $\frac{((1+x+x^2)(x-1))^n}{x^n} = \frac{(x^3-1)^n}{x^n}$.

Now, expand $(x^3-1)^n$ using the binomial theorem: $(x^3-1)^n = \sum_{k=0}^n {}^nC_k (x^3)^k (-1)^{n-k} = \sum_{k=0}^n (-1)^{n-k} {}^nC_k x^{3k}$.

So, $\frac{(x^3-1)^n}{x^n} = \frac{\sum_{k=0}^n (-1)^{n-k} {}^nC_k x^{3k}}{x^n} = \sum_{k=0}^n (-1)^{n-k} {}^nC_k x^{3k-n}$.

The constant term in this expansion is obtained when the exponent of $x$ is 0, i.e., $3k-n=0$, or $3k=n$. For the constant term to exist, $n$ must be a multiple of 3. If $n$ is a multiple of 3, let $n=3m$ for some integer $m$. Then $3k=3m$, which gives $k=m$. Since $0 \le k \le n$, we must have $0 \le m \le 3m$, which is true for $m \ge 0$. The value of $k$ is $n/3$. The constant term is $(-1)^{n-n/3} {}^nC_{n/3}$. If $n=3m$, the constant term is $(-1)^{3m-m} {}^nC_m = (-1)^{2m} {}^nC_m = ((-1)^2)^m {}^nC_m = 1^m {}^nC_m = {}^nC_m$. Since $m=n/3$, the constant term is ${}^nC_{n/3}$.

If $n$ is not a multiple of 3, there is no integer $k$ in the range $0 \le k \le n$ such that $3k=n$. In this case, the coefficient of $x^0$ is 0.

So, the sum $S = \sum_{r=0}^{n} (-1)^r a_r {}^nC_r$ is equal to:

• ${}^nC_{n/3}$ if $n \equiv 0 \pmod 3$.

• 0 if $n \not\equiv 0 \pmod 3$.

Now let's check the given options:

(A) 0 if $n=57$. $57 = 3 \times 19$. $n \equiv 0 \pmod 3$. The sum is ${}^{57}C_{19}$. So this option is incorrect.

(B) 0 if $n=77$. $77 = 3 \times 25 + 2$. $n \not\equiv 0 \pmod 3$. The sum is 0. So this option is correct.

(C) ${}^{24}C_8$ if $n=24$. $24 = 3 \times 8$. $n \equiv 0 \pmod 3$. The sum is ${}^{24}C_{24/3} = {}^{24}C_8$. So this option is correct.

(D) ${}^{39}C_{13}$ if $n=39$. $39 = 3 \times 13$. $n \equiv 0 \pmod 3$. The sum is ${}^{39}C_{39/3} = {}^{39}C_{13}$. So this option is correct.

Since the question asks which of the options is equal to the sum, and multiple options are correct, this is a multiple correct options question.