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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

Answers (1)
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A admin

 

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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