Question

The sum of an infinite gp with positive terms is 48 and the sum of first two terms is 36 then find the 22nd term is ..
answer pls...
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Answer 1

Answer:-

Let the first term of the GP be a and common ratio be r.

Given:-

Sum of infinite GP (S∞) = 48

We know that,

Sum of an infinite GP = a/1 - r

So,

⟹ a/1 - r = 48

⟹ a = 48(1 - r)

⟹ a = 48 - 48r -- equation (1)

Also given that,

Sum of first two terms = 36

First two terms of a GP are a, ar.

⟹ a + ar = 36

Substitute the value of a from equation (1).

⟹ 48 - 48r + (48 - 48r)r = 36

⟹ 48 - 48r + 48r - 48r² = 36

⟹ 48 - 36 = 48r²

⟹ 12 = 48r²

⟹ 12/48 = r²

⟹ r² = 1/4

⟹ r = ½

* Substitute r = ½ in equation (1)

⟹ a = 48 - 48(1/2)

⟹ a = 48 - 24

⟹ a = 24

Now,

We know;

nth term of a GP = a × rⁿ⁻¹

⟹ a₂₂ = a × r²²⁻¹

⟹ a₂₂ = a × r²¹

⟹ a₂₂ = 24 × (½)²¹

⟹ a₂₂ = 2³ × 3 × 2⁻²¹ [ ∵ 1/aⁿ = a⁻ⁿ ]

⟹ a₂₂ = 3 × 2³⁻²¹ [ ∵ aᵐ × aⁿ = aᵐ⁺ⁿ ]

⟹ a₂₂ = 3 × 2⁻¹⁸

∴ The 22nd term of the given GP is 3 × 2⁻¹⁸.