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# Help me answer: - Letbe in GP withforand S be the set of pairs (r, k), r, k( the set of natural numbers) for whi - Sequence and series - JEE Main

Let $a_1, a_2, a_3, ..., a_{10}$ be in GP with $a_i > 0$ for $i = 1,2,..., 10$ and S be the set of pairs (r, k), r, k$\in N$( the set of natural numbers) for which

$\begin{vmatrix} \log_e a_1^ra_2^k & \log_e a_2^ra_3^k &\log_e a_3^ra_4^k \\ \log_e a_4^ra_5^k & \log_e a_5^ra_6^k & \log_e a_6^ra_7^k \\ \log_e a_7^ra_8^k &\log_e a_8^ra_9^k &\log_e a_9^ra_{10}^k \end{vmatrix} = 0$

Then the number of elements in S, is:

• Option 1)

4

• Option 2)

Infinitely many

• Option 3)

10

• Option 4)

2

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Geometric Progession (GP) -

A progression of non - zero terms, in which every term bears to the preceding term a constant ratio.

- wherein

eg 2, 4, 8, 16,- - - - - -

and

100, 10, 1, 1/10,- - - - - - -

General term of a GP -

$T_{n}= ar^{n-1}$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

$D=\begin{vmatrix} \log_e a_1^ra_2^k & \log_e a_2^ra_3^k &\log_e a_3^ra_4^k \\ \log_e a_4^ra_5^k & \log_e a_5^ra_6^k & \log_e a_6^ra_7^k \\ \log_e a_7^ra_8^k &\log_e a_8^ra_9^k &\log_e a_9^ra_{10}^k \end{vmatrix}$

Apply coloumn operation

$C_{3}\rightarrow C_{3}-C_{2}$

$C_{2}\rightarrow C_{2}-C_{1}$

we get D = 0

OR

$a_{1},a_{2},a_{3}...............,a_{10}$   are in G.P.

assume $a_{1}=1,a_{2}=1,a_{3}=1...............,a_{10}=1$

Since $1,1,1,......................$ are in G.P. with common ratio 1

So, $\log (1)=0$

Value of D become 0.

Option 1)

4

Option 2)

Infinitely many

Option 3)

10

Option 4)

2

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