Question

if n > 2, then evaluate
$\begin{gathered}\displaystyle\sf\dfrac{\sum\limits_{r=2}^{n} (-2)^r \left| \begin{array}{ccc} \sf {}^{n-2} C_{r-2} &\sf {}^{n-2} C_{r-1} &\sf {}^{n-2} C_r \\\\\sf -3&\sf 1 &\sf 1\\\\\sf -3&\sf 1&\sf 1\\\\\sf 2&\sf -1&\sf 0 \end{array}\right| +(-1)^{n+1}}{2n-1}\end{gathered}$
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Answer 1

AnSwEr:—

$\begin{gathered}\small\rm \Delta_{\sf r} \sf = \left| \begin{array}{ccc} \sf {}^{n-2} C_{r-2} &\sf {}^{n-2} C_{r-1} &\sf {}^{n-2} C_r \\\\\sf -3&\sf 1 &\sf 1\\\\\sf 2&\sf -1&\sf 0 \end{array}\right| \: , \: r=2,3,\dots ,n\end{gathered} </p><p>$

Applying,

$</u></strong></p><p><strong><u>[tex]\sf c_1 \to c_1+2c_2+c_3</u></strong></p><p><strong><u>[tex]\sf c_1 \to c_1+2c_2+c_3$

$\begin{gathered}\scriptsize\displaystyle\sf = \displaystyle\sf\sum\limits_{r=2}^{n} (-2)^r \: \left|\begin{array}{ccc} \sf {}^{n+2} C_{r-2} + {}^{n-1}C_{r-1} &\sf {}^{n-2} C_{r-1} &\sf {}^{n-2} C_r \\\\\sf 0&\sf 0&\sf 1\\\\\sf 0&\sf -1&\sf 0\end{array}\right|\end{gathered}$

Applying,

$\sf C_1\to C_1 +2C_2+C_3C$

$\begin{gathered}\footnotesize\displaystyle\sf = \sum_{r=2}^{n} (-2)^r \left| \begin{array}{ccc} \sf {}^{n-1}C_{r-1}+{}^{n-1}C_r &\sf {}^{n-2}C_{r-1}&\sf {}^{n-2} C_r \\\\\sf 0&\sf 1&\sf 1\\\\\sf 0&\sf -1&\sf 0 \end{array}\right|\end{gathered} </p><p>$

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