Prove that: (a+1)(a+2)&(a+2)&1 a+2)(a+3)&(a+3)&1 a+3)(a+4)&(a+4) … - Vidyadip

Question

Prove that: $\begin{vmatrix}(a+1)(a+2)&(a+2)&1\\(a+2)(a+3)&(a+3)&1\\(a+3)(a+4)&(a+4) &1\end{vmatrix}=-2$

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Answer

$\begin{vmatrix}(a+1)(a+2)&a+2&1\\(a+2)(a+3)&a+3&1\\(a+3)(a+4)&a+4 &1\end{vmatrix}=-2$
$\text{L.H.S}=\begin{vmatrix}(a+1)(a+2)&a+2&1\\(a+2)(a+3)&a+3&1\\(a+3)(a+4)&a+4 &1\end{vmatrix}$ Apply R3 → R3 - R2
$=\begin{vmatrix}(a+1)(a+2)&a+2&1\\(a+2)(a+3)&a+3&1\\(a+3)2&1&0\end{vmatrix}$ Apply R2 → R2 - R1
$=\begin{vmatrix}(a+1)(a+2)&a+2&1\\(a+2)2&1&0\\(a+3)2&1&0\end{vmatrix}$
$=[(2a+4)(1)-(1)(2a+6)]$
$=-2$
$=\text{R.H.S}$

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