Multiple choice

The sum of the series $1\cdot { 3 }^{ 2 }+2\cdot { 5 }^{ 2 }+3\cdot { 7 }^{ 2 }+...$ upto $20$ terms is

  1. $188090$
  2. $189080$
  3. $199080$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The n-th term is n * (2n+1)^2 = n * (4n^2 + 4n + 1) = 4n^3 + 4n^2 + n. Sum = 4*sum(n^3) + 4*sum(n^2) + sum(n). For n=20: 4*(20*21/2)^2 + 4*(20*21*41/6) + (20*21/2) = 4*(44100) + 4*(2870) + 210 = 176400 + 11480 + 210 = 188090.

AI explanation

Observe that the coefficients of the squared terms form an arithmetic progression 1, 2, 3, while the bases form an arithmetic progression 3, 5, 7. The nth term of the series is given by n multiplied by (2n + 1)^2, which expands to 4n^3 + 4n^2 + n. Apply the standard summation formulas for squares and cubes: the sum of n squares is n(n+1)(2n+1)/6 and the sum of n cubes is (n(n+1)/2)^2. Substituting n = 20 into 4(n(n+1)/2)^2 + 4(n(n+1)(2n+1)/6) + n(n+1)/2 results in 188090.

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