text: # Why is $(1+2+3+4+…+n)^2$ equal to $1^3+2^3+3^3…+n^3$? [duplicate]
  
  I noticed that the sum of the first $n$ cubes is equal to the square of sum of the first $n$ natural numbers: $$\sum\limits_{i=1}^n i^3=\frac{n^2(n+1)^2}{4}=\left(\frac{n(n+1)}{2}\right)^2=\left(\sum\limits_{i=1}^n i\right)^2$$
  
  Is there a clever way to explain (not prove) this identity?
explanation: The text extract contains a math question about why the square of the sum of the first \( n \) natural numbers is equal to the sum of the first \( n \) cubes. The user also provides an identity and asks for a clever way to explain this identity.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Why is $(1+2+3+4+…+n)^2$ equal to $1^3+2^3+3^3…+n^3$? [duplicate]
  
  I noticed that the sum of the first $n$ cubes is equal to the square of sum of the first $n$ natural numbers: $$\sum\limits_{i=1}^n i^3=\frac{n^2(n+1)^2}{4}=\left(\frac{n(n+1)}{2}\right)^2=\left(\sum\limits_{i=1}^n i\right)^2$$
  
  Is there a clever way to explain (not prove) this identity?
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math question about why the square of the sum of the first \( n \) natural numbers is equal to the sum of the first \( n \) cubes. The user also provides an identity and asks for a clever way to explain this identity.
  
  Label: question