https://file.aiquickdraw.com/mathbot/file/1726677886897cq6nkjmv.jpg

The equation in the image provided is:

\[
2 + 6 + 12 + \dots + n(n+1) = n(n+2), \quad n \geq 1.
\]

This represents a sum of multiples, likely based on an arithmetic or quadratic pattern.

To clarify:

- The left-hand side seems to be the sum of terms, where the general form for the \(k\)-th term appears to be \(k(k+1)\), which begins with \(2(2+1) = 6\), followed by \(3(3+1) = 12\), and so on until \(n(n+1)\).
  
The right-hand side is \(n(n+2)\), which simplifies to a quadratic expression. 

Let's confirm and analyze the relationship:

1. **Sum pattern**: The left-hand side is a sum of terms of the form \(2 + 6 + 12 + \cdots + n(n+1)\).
2. **Target identity**: It is to be shown that this sum equals the quadratic expression \(n(n+2)\).

Would you like me to go through the detailed steps to prove this identity?

2 + 6 + 12 + ... + n(n+1) = n(n+2), n ≥ 1

Explore the summation of a series involving multiples of n and the corresponding quadratic identity n(n+2). This math problem is suitable for high school students learning algebraic manipulation and summation of arithmetic series.

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