# 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?
The text passage contains a mathematical question about the relationship between the square of the sum of the first \( n \) natural numbers and the sum of the first \( n \) cubes. It also provides an identity that explains this relationship.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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?
user
The text passage contains a mathematical question about the relationship between the square of the sum of the first \( n \) natural numbers and the sum of the first \( n \) cubes. It also provides an identity that explains this relationship.

TYPE: SINGLE_QUESTION_WITH_ANSWER