Question

Let the 13th term in the expansion of $(x^2 + \frac{2}{x})^n$ is independent of $x$. If the sum of the positive divisors of $n$ is $k$, then the value of $\frac{k}{3}$ is

• A. 13
• B. 39
• C. 18
• D. 12

Answer 1

Answer: (A)

13

Answer 2

The general term in the expansion of $(x^2 + \frac{2}{x})^n$ is $T_{r+1} = \binom{n}{r} (x^2)^{n-r} (\frac{2}{x})^r$. The exponent of $x$ is $2(n-r) - r = 2n - 3r$. The 13th term means $r=12$. For the term to be independent of $x$, $2n - 3r = 0$. Substituting $r=12$, we get $2n - 3(12) = 0$, so $2n = 36$, which gives $n=18$. The sum of the positive divisors of $n=18$ is $k$. The divisors of $18 = 2^1 \cdot 3^2$ are $1, 2, 3, 6, 9, 18$. The sum $k = 1+2+3+6+9+18 = 39$. The value of $\frac{k}{3}$ is $\frac{39}{3} = 13$.