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Mathematics

Assertion (A): The 50th term from the end of the G.P. 4, 6, 9, 272,......656164\dfrac{27}{2}, ……\dfrac{6561}{64} is 272\dfrac{27}{2}.

Reason (R): nth term from the end of a G.P. is given by lrn−1\dfrac{l}{r^{n-1}}.

  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

G.P.

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Answer

Given,

a = 4

n = 50

r = 64=32\dfrac{6}{4} = \dfrac{3}{2}

l = 656164\dfrac{6561}{64}

We know that,

nth term from end=lrn−1\text{nth term from end} = \dfrac{l}{r^{n - 1}}

Substituting values we get,

⇒50th term from end=656164(32)50−1=3826349249=3826×249349=249−6×38−49=243×3−41\Rightarrow \text{50th term from end} = \dfrac{\dfrac{6561}{64}}{\Big(\dfrac{3}{2}\Big)^{50-1}} \\[1em] = \dfrac{\dfrac{3^8}{2^6}}{\dfrac{3^{49}}{2^{49}}} \\[1em] = \dfrac{3^8}{2^6} \times \dfrac{2^{49}}{3^{49}} \\[1em] = 2^{49 - 6} \times 3 ^{8 - 49} \\[1em] = 2^{43} \times 3^{-41}

= 243341\dfrac{2^{43}}{3^{41}} ≠ 272\dfrac{27}{2}.

So, Assertion is false.

nth term from the end of a G.P. is given by: lrn−1\dfrac{l}{r^{n-1}}

So, reason is true.

Hence, option 2 is the correct option.

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