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Mathematics

Assertion (A) : Product of 5th term from beginning and 5th term from end of the G.P. : 3 - 6 + 12 - 24 + ….. -384 is -1152.

Reason (R) : In G.P. product of nth term from beginning and nth term from end is 1st term × last term.

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true.

  4. Both A and R are false.

AP GP

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Answer

G.P. : 3 - 6 + 12 - 24 + ….. -384

Common ratio (r) = −63\dfrac{-6}{3} = -2.

Let no. of terms be n.

Last term = -384

⇒ -384 = arn - 1

⇒ -384 = 3(-2)n - 1

⇒ (-2)n - 1 = −3843-\dfrac{384}{3}

⇒ (-2)n - 1 = -128

⇒ (-2)n - 1 = (-2)7

⇒ n - 1 = 7

⇒ n = 7 + 1 = 8.

5th term beginning = a5

= ar5 - 1

= ar4

= 3(-2)4

= 3 × 16

= 48.

5th term from end = (8 - 5 + 1)th term from beginning

= 4th term from beginning

= ar4 - 1

= ar3

= 3(-2)3

= 3 × -8

= -24.

5th term beginning × 5th term from end

= 48 × -24

= -1152.

∴ Assertion (A) is true.

Suppose there are n terms in a G.P.

Then, nth term from end = 1st term.

Thus,

In G.P. product of nth term from beginning and nth term from end is 1st term × last term.

∴ Reason (R) is true.

Hence, Option 3 is the correct option.

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