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Black Advanced Problems in Mathematics
Exam
Black Advanced Problems in Mathematics
Subject
Mathematics
Chapter
Sequences and Series

Black Advanced Problems in Mathematics · Mathematics · Sequences and Series

If S_1, S_2 and S_3 are the sums of first n natural numbers, their...

If S1,S2S_1, S_2 and S3S_3 are the sums of first nn natural numbers, their squares and their cubes respectively, then S14S22−S22S32S12+S32=\frac{S_1^4 S_2^2 - S_2^2 S_3^2}{S_1^2 + S_3^2} =

Options

  1. A44
  2. B22
  3. C11
  4. D00

Correct answer

4

Step-by-step solution

We know the standard formulas: S1=n(n+1)2S2=n(n+1)(2n+1)6S3=S_1 = \frac{n(n+1)}{2}S_2 = \frac{n(n+1)(2n+1)}{6}S_3 =\left(\frac{n(n+1)}{2}\right)^2NoticethatNotice thatS_3 = S_1^2.Substitute. SubstituteS_3 = S_1^2intothenumerator:into the numerator:\frac{S_1^4 S_2^2 - S_2^2 (S_1^2)^2}{S_1^2 + S_3^2} = \frac{S_1^4 S_2^2 - S_2^2 S_1^4}{S_1^2 + S_3^2} = \frac{0}{S_1^2 + S_3^2} = 0.
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