If a1- a2- a3 -are in AP- then the value of - a1 a2 1- a2 a3 1- a3 a4 1 - is

The value of [[ a_1 a_2 1; a_2 a_3 1; a_3 a_4 1 ]] isGiven a1, a2, a3 ……are in APa_2 a_1=a_3 a_2=a_4 a_3=....=d0ex0exLet A =[ a_1 a_2 1; a_2 a_3 1; a_ ...

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Standard VIII

Mathematics

Rectangle

If a1, a2, a3...

Question

If $a 1, a 2, a 3 \dots \dots$ are in AP, then the value of $[\begin{matrix} a_{1} & a_{2} & 1 \\ a_{2} & a_{3} & 1 \\ a_{3} & a_{4} & 1 \end{matrix}]$ is

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Solution

The value of $[\begin{matrix} a_{1} & a_{2} & 1 \\ a_{2} & a_{3} & 1 \\ a_{3} & a_{4} & 1 \end{matrix}]$ is
Given $a 1, a 2, a 3 \dots \dots$ are in AP
$a_{2} - a_{1} = a_{3} - a_{2} = a_{4} - a_{3} = . . . . = d L e t A = \begin{matrix} a_{1} & a_{2} & 1 \\ a_{2} & a_{3} & 1 \\ a_{3} & a_{4} & 1 \end{matrix}$
On Performing
$R_{2} \to R_{2} - R_{1} a n d R_{3} \to R_{3} - R_{2}, w e g e t A = \begin{matrix} a_{1} & a_{2} & 1 \\ a_{2} - a_{1} & a_{3} - a_{2} & 0 \\ a_{3} - a_{2} & a_{4} - a_{3} & 0 \end{matrix} = \begin{matrix} a_{1} & a_{2} & 1 \\ d & d & 0 \\ d & d & 0 \end{matrix} = 0, S i n c e t w o r o w s a r e i d e n t i c a l$
Hence the value of $A = [\begin{matrix} a_{1} & a_{2} & 1 \\ a_{2} & a_{3} & 1 \\ a_{3} & a_{4} & 1 \end{matrix}] i s 0$

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If a1,a2,a3……are in AP, then the value of a1a21a2a31a3a41 is

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If $a 1, a 2, a 3 \dots \dots$ are in AP, then the value of $[\begin{matrix} a_{1} & a_{2} & 1 \\ a_{2} & a_{3} & 1 \\ a_{3} & a_{4} & 1 \end{matrix}]$ is