# When is 1+2+ +n a perfect square?

• CornMuffin
In summary, the formula for determining when 1+2+...+n is a perfect square is: n(n+1)/2 = m^2, where n is the number of terms and m is the perfect square. To prove this, you can use mathematical induction. There are no values of n for which 1+2+...+n is not a perfect square, and examples include n=4 and n=10. This concept has real-life applications in various mathematical and engineering problems, such as determining the number of squares in a grid pattern or finding the sum of consecutive odd or even numbers.
CornMuffin

## Homework Statement

Find the values of $$n\geq 1$$ for which 1! + 2! + ... + n! is a perfect square in the integers.

## The Attempt at a Solution

n=1 and n=3 works, but I don't know how to find anymore, or prove that there aren't anymore

Petek said:

well I proved it, thanks

## 1. What is the formula for determining when 1+2+...+n is a perfect square?

The formula for determining when 1+2+...+n is a perfect square is: n(n+1)/2 = m^2, where n is the number of terms and m is the perfect square.

## 2. How can I prove that 1+2+...+n is a perfect square?

To prove that 1+2+...+n is a perfect square, you can use mathematical induction. First, show that the formula n(n+1)/2 is true for n=1. Then, assume it is true for n=k and prove it is true for n=k+1. This will prove that the formula holds for all natural numbers and therefore, 1+2+...+n is a perfect square.

## 3. Are there any values of n for which 1+2+...+n is not a perfect square?

No, there are no values of n for which 1+2+...+n is not a perfect square. This is because the formula n(n+1)/2 will always result in a perfect square number for any value of n.

## 4. Can you provide an example of when 1+2+...+n is a perfect square?

Yes, for n=4, 1+2+3+4 = 10 which is a perfect square (3^2). Similarly, for n=10, 1+2+...+10 = 55 which is a perfect square (10^2).

## 5. Is there a real-life application of this concept?

Yes, this concept is used in various mathematical and engineering problems. For example, it can be used to determine the number of squares that can be formed in a grid pattern or to find the sum of consecutive odd or even numbers.

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