Multiple choice

The 7th term of the series $1+\cfrac { 1 }{ \left( 1+3 \right) } { \left( 1+2 \right) }^{ 2 }++\cfrac { 1 }{ \left( 1+3+5 \right) } { \left( 1+2+3 \right) }^{ 2 }+....$ is equal to

  1. $16$
  2. $18$
  3. $20$
  4. $22$
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A Correct answer
Explanation

The nth term follows the pattern (1 + 2 + ... + n)^2 divided by (1 + 3 + ... + (2n - 1)). These sums are n(n + 1)/2 and n^2 respectively, so the nth term is (n + 1)^2/4, giving 16 for n = 7.

AI explanation

For the 7th term, the multiplier is the square of the sum of the first 7 natural numbers, which is (1+2+3+4+5+6+7)^2 = (28)^2 = 784. The denominator is the sum of the first 7 odd numbers, which is 7^2 = 49. Dividing the two parts gives 784 / 49 = 16.

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