If a1, a2, …. . are in A P and Δ=| an an+2 an+4 an+6 an+8 an+10 an+12

Name: Problem 138, Chapter 2: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-08-18T08:35:54-08:00
Duration: 2 min 8 s
Channel: Vikash Ranjan
Description: If a1, a2, …. . are in A P and Δ=| an an+2 an+4 an+6 an+8 an+10 an+12 an+14 an+16 | is (a) an+an+8+an+16 (b) an an+8 an+16 (c) 0 (d) 1

step 1 In the problem given that A1, A2 are in AP, that is a thematic progression. So we have A .I that

Question

If $a_{1}, a_{2}, \ldots . .$ are in $A P$ and $\Delta=\left|\begin{array}{ccc}a_{n} & a_{n+2} & a_{n+4} \\ a_{n+6} & a_{n+8} & a_{n+10} \\ a_{n+12} & a_{n+14} & a_{n+16}\end{array}\right|$ is (a) $a_{n}+a_{n+8}+a_{n+16}$ (b) $a_{n} a_{n+8} a_{n+16}$ (c) 0 (d) 1

   If $a_{1}, a_{2}, \ldots . .$ are in $A P$ and $\Delta=\left|\begin{array}{ccc}a_{n} & a_{n+2} & a_{n+4} \\ a_{n+6} & a_{n+8} & a_{n+10} \\ a_{n+12} & a_{n+14} & a_{n+16}\end{array}\right|$ is
(a) $a_{n}+a_{n+8}+a_{n+16}$
(b) $a_{n} a_{n+8} a_{n+16}$
(c) 0
(d) 1

IIT JEE Super Course in Mathematics ALGEBRA II

Trishna Knowledge… 2011 Edition

Chapter 2, Problem 138

Instant Answer

Solved by Expert Vikash Ranjan

08/18/2021

Step 1

This can be written as $a_{i} = a_{j} + (i - j)d$ where $d$ is the common difference and $i, j$ are any two indices in the sequence. Show more…

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If $a_{1}, a_{2}, \ldots . .$ are in $A P$ and $\Delta=\left|\begin{array}{ccc}a_{n} & a_{n+2} & a_{n+4} \\ a_{n+6} & a_{n+8} & a_{n+10} \\ a_{n+12} & a_{n+14} & a_{n+16}\end{array}\right|$ is (a) $a_{n}+a_{n+8}+a_{n+16}$ (b) $a_{n} a_{n+8} a_{n+16}$ (c) 0 (d) 1

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Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This property implies that each term can be expressed as a linear function of its position in the sequence, which simplifies handling expressions involving these terms.

Determinant Properties

Determinants are scalar values computed from square matrices that encapsulate properties such as linearity. One key property of determinants is that if any row (or column) is a linear combination of other rows (or columns), the determinant of the matrix is zero.

Linear Dependence in Rows or Columns

When a matrix is constructed with elements from an arithmetic progression, the linearity of the terms can lead to rows or columns that are linearly dependent. This means that one row (or column) can be expressed as a combination of others, resulting in a determinant value of zero.

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