Multiple choice

The sum of the series $\dfrac { 9 }{ 5^{ { 2 } }\cdot 2.1 } +\dfrac { 13 }{ 5^{ { 3 } }\cdot 3.2 } +\dfrac { 17 }{ 5^{ { 4 } }\cdot 4.3 } ..........$ Infinite terms

  1. $\dfrac { 1 } { 25 }$
  2. $\dfrac { 1 } { 5 }$

  3. Not a finite number
  4. $\dfrac { 9 } { 5 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The general term is (4n+5) / (5^(n+1) * n(n+1)). This can be decomposed using partial fractions. The sum telescopes to 1/5.

AI explanation

The general term of the series can be written by observing the patterns in the numerators and denominators, where the numerator follows the progression 9, 13, 17, which is 4n plus 5. The denominators are 5 to the power of (n plus 1) multiplied by the product of (n plus 1) and n. Rewriting the general term T_n as (4n plus 5) divided by ((n + 1) times n times 5 to the power of (n + 1)), we separate it into partial fractions to telescope the sum. Multiplying the numerator and denominator by 5 gives (20n plus 25) divided by (5 to the power of (n + 1) times (n + 1) times n), which simplifies to the difference between 1 divided by (n times 5 to the power of n) and 1 divided by ((n + 1) times 5 to the power of (n + 1)). Summing this from n equals 1 to infinity causes all intermediate terms to cancel out, leaving only the first term, which is 1 divided by 5.

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