Multiple choice

If the sum of the first ten terms of the series $\left (1 \dfrac {3}{5}\right )^{2} + \left (2 \dfrac {2}{5}\right )^{2} + \left (3 \dfrac {1}{5}\right )^{2} + 4^{2} + \left (4 \dfrac {4}{5}\right )^{2} + ..............$ is $\dfrac {16}{5}m$, then $m$ is equal to:

  1. $102$
  2. $101$
  3. $100$
  4. $99$
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B Correct answer
Explanation

Series terms: (8/5)^2, (12/5)^2, (16/5)^2, (20/5)^2... This is (4/5)^2 * (2^2, 3^2, 4^2, 5^2...). The sum of the first 10 terms is (16/25) * sum of squares from 2 to 11. Sum of squares 1^2 to 11^2 = (11*12*23)/6 = 506. Subtract 1^2 = 505. Sum = (16/25) * 505 = 16 * 20.2 = 323.2. 323.2 = (16/5) * m. m = 323.2 * 5 / 16 = 101.

AI explanation

Convert the mixed fractions to improper fractions to see that the terms follow the sequence (8/5)^2, (12/5)^2, (16/5)^2, forming an arithmetic progression with a first term of 8/5 and a common difference of 4/5. The sum of the squares of the first ten terms of this sequence is 1/25 multiplied by the sum of (8 + 4(n-1))^2 from 1 to 10. Using the formulas for the sum of n squares and the sum of n natural numbers, the total evaluates to 1616/5. Setting this equal to 16/5 * m gives m = 101.

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