Find the sum of the G.P.:

2/5, 2/25, 2/125, 2/625,... to n terms. 

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UP Police SI (दरोगा) Official PYP (Held On: 12 Nov 2021 Shift 2)
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  1. 5/4(1-(1/5n))
  2. 2/5(1-(1/5n​))
  3. 1/2(1-(1/5n​))
  4. 4/5(1-(1/5n​))

Answer (Detailed Solution Below)

Option 3 : 1/2(1-(1/5n​))
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Detailed Solution

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Given:

2/5, 2/25, 2/125, 2/625,... to n terms.

Formula used:

Sum of a G.P. to n terms = a(\(\dfrac{1-r^n}{1-r}\))

Where,

a = first term

r = common ratio

Calculation:

Here, a = 2/5

r = 2/25 ÷ 2/5 = 1/5

Sum to n terms = a\(\dfrac{1-r^n}{1-r}\)

⇒ Sum = (2/5)\(\dfrac{1-(1/5)^n}{1-1/5}\)

⇒ Sum = (2/5)\(\dfrac{1-(1/5)^n}{4/5}\)

⇒ Sum = (2/5) × (5/4) × [1 - (1/5)n]

⇒ Sum = 1/2 × [1 - (1/5)n]

∴ The correct answer is option 3.

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