If ae, am, an are the ℓth m th and n th terms of an AP, then | at ℓ 1 am m

step 1 So in the given question we are told that the elements A L, A. M and A .N are the Lth they are t

If ae, am, an are the ℓth m th and n th terms of an AP, then...

If $a_{e}, a_{m}, a_{n}$ are the $\ell$ th $m$ th and $n$ th terms of an AP, then $\left|\begin{array}{lll}a_{t} & \ell & 1 \\ a_{m} & m & 1 \\ a_{n} & n & 1\end{array}\right|$ is equal to
(a) $\ell \mathrm{mn}$
(b) $\ell+\mathrm{m}+\mathrm{n}$
(c) $\ell+\mathrm{m}+\mathrm{n}-3$
(d) 0

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Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence in which each term after the first is obtained by adding a fixed constant, called the common difference, to the previous term. The nth term of an AP can be expressed as a linear function of n, making the relationship between the term’s value and its position a straight line when graphed.

Determinant

The determinant of a square matrix is a scalar value that can be computed from its elements and provides important information about the matrix, such as whether it is invertible. In the context of a 3x3 matrix, the determinant can be used to verify properties like linear dependence of rows or columns.

Collinearity Test

A common application of determinants in geometry is the collinearity test. When three points are represented in the form (x, y, 1) in a matrix, the determinant of that matrix equals zero if and only if the points lie on a single straight line. This property is key to understanding why, in problems involving arithmetic progressions (where the terms form a linear sequence), certain determinants simplify to zero.

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