Suppose det [ k = 0 n k k = 0 n C k n k 2 k = 0 n C k n k k = 0 n C k n 3 k ] = 0 holds for some positive integer n . The k = 0 n C k n k + 1 equals _________ . [2019]

Question

Suppose det $[\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} k^{2} \\ \sum_{k = 0}^{n} {}^{n}{C_{k}} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} 3^{k} \end{matrix}] = 0$ holds for some positive integer $n$ . The $\sum_{k = 0}^{n} \frac{{}^{n}{C_{k}}}{k + 1}$ equals _________ . [2019]

Smart Achievers · Accepted Answer

(6.20)

$Here \sum_{k = 0}^{n} k = \frac{n (n + 1)}{2}$

$\sum_{k = 0}^{n} {}^{n}C_{k} k^{2} = \sum_{k = 1}^{n} \frac{n}{k} . {}^{n - 1}C_{k - 1} . k^{2} = \sum_{k = 1}^{n} n . {}^{n - 1}C_{k - 1} . k$

$= n \sum_{k = 1}^{n} {}^{n - 1}C_{k - 1} (k - 1 + 1)$ $= n [\sum_{k = 2}^{n} \frac{n - 1}{k - 1} {}^{n - 2}C_{k - 2} (k - 1) + \sum_{k = 1}^{n} {}^{n - 1}C_{k - 1}]$

$= n (n - 1) 2^{n - 2} + n \times 2^{n - 1}$

$\sum_{k = 0}^{n} {}^{n}C_{k} \times k = \sum_{k = 1}^{n} \frac{n}{k} {}^{n - 1}C_{k - 1} \times k = n \times 2^{n - 1}$

$and \sum_{k = 0}^{n} {}^{n}C_{k} 3^{k} = 4^{n}$

$∴$ det $[\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}C_{k} k^{2} \\ \sum_{k = 0}^{n} {}^{n}C_{k} k & \sum_{k = 0}^{n} {}^{n}C_{k} 3^{k} \end{matrix}] = 0$

$\Rightarrow$ $| \begin{matrix} \frac{n (n + 1)}{2} & n (n - 1) 2^{n - 2} + n \times 2^{n - 1} \\ n \times 2^{n - 1} & 4^{n} \end{matrix} |$ $= 0$

$\Rightarrow n (n + 1) \times 2^{2 n - 1} - n^{2} [(n - 1) 2^{2 n - 3} + 2^{2 n - 2}] = 0$

$\Rightarrow 2^{2 n - 3} \times n [4 (n + 1) - n (n - 1 + 2)] = 0$

$\Rightarrow 2^{2 n - 3} \times n [4 n + 4 - n^{2} - n] = 0$

$\Rightarrow n^{2} - 3 n - 4 = 0 \Rightarrow n = 4$

$∴$ $\sum_{k = 0}^{n} \frac{{}^{n}C_{k}}{k + 1} = \sum_{k = 0}^{4} \frac{{}^{4}C_{k}}{k + 1} = \frac{{}^{4}C_{0}}{1} + \frac{{}^{4}C_{1}}{2} + \frac{{}^{4}C_{2}}{3} + \frac{{}^{4}C_{3}}{4} + \frac{{}^{4}C_{4}}{5}$

$= 1 + 2 + 2 + 1 + \frac{1}{5} = 6.20$

Solution

Q.
Suppose det $[\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} k^{2} \\ \sum_{k = 0}^{n} {}^{n}{C_{k}} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} 3^{k} \end{matrix}] = 0$ holds for some positive integer $n$ . The $\sum_{k = 0}^{n} \frac{{}^{n}{C_{k}}}{k + 1}$ equals _________ . [2019]

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Solution

Q. Suppose det [∑k=0nk∑k=0nCknk2∑k=0nCknk∑k=0nCkn3k]=0 holds for some positive integer n. The ∑k=0nCknk+1 equals _________ . [2019]

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Q.
Suppose det $[\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} k^{2} \\ \sum_{k = 0}^{n} {}^{n}{C_{k}} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} 3^{k} \end{matrix}] = 0$ holds for some positive integer $n$ . The $\sum_{k = 0}^{n} \frac{{}^{n}{C_{k}}}{k + 1}$ equals _________ . [2019]