Multiple choice

$n^{th}$ term of the series $1+\dfrac { 4 }{ 5 } +\dfrac { 7 }{ { 5 }^{ 2 } } +\dfrac { 10 }{ { 5 }^{ 3 } } +..........................$

  1. $\dfrac { 3n+1 }{ { 5 }^{ n-1 } } $
  2. $\dfrac { 3n-1 }{ { 5 }^{ n } } $
  3. $\dfrac { 3n-2 }{ { 5 }^{ n-1 } } $
  4. $\dfrac { 3n+2 }{ { 5 }^{ n-1 } } $
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C Correct answer
Explanation

The numerators form an arithmetic progression 1, 4, 7, 10... with nth term 3n-2. The denominators form a geometric progression 5^0, 5^1, 5^2... with nth term 5^(n-1). Combining these gives (3n-2) / 5^(n-1).

AI explanation

Examine the pattern in the numerators: 1, 4, 7, 10 forms an arithmetic progression with a common difference of 3. The formula for the numerator of the nth term is 1 + (n-1)3, which simplifies to 3n - 2. The denominators follow the pattern 5^0, 5^1, 5^2, 5^3, meaning the denominator for the nth term is 5^(n-1). Combining these gives the nth term as (3n-2) / 5^(n-1).

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