# Arithmetic Series Perfect Square

• Frillth
In summary, the only way to make a sequence of integers with the property that the sum of the first n terms is a perfect square for all integers n is to have a sequence where adding a successive term makes the sum equal the successive square. This can be achieved by setting a_1 to be a perfect square. Other sequences can be found by setting a_1 to be any integer multiplied by x^2, where x is any integer.
Frillth

## Homework Statement

I need to find all arithmetic sequences of integers with the property that the sum of the first n terms is a perfect square for all integers n.

## Homework Equations

a_n = nth term of the sequence = a_1 + (n-1)d
d = common difference
Sum of the first n terms of the sequence = n[2a_1+(n-1)d]/2

## The Attempt at a Solution

I know that the sequence 1, 3, 5, 7, 9... sums to a perfect square every time, as will 1x^2, 3x^2, 5x^2..., with x being any integer.

This is the only way I can find to make the sequence sum to a perfect square every time. If this isn't the only way, what are the others? If this is, how can I prove it is the only way?

well you noticed that $\sum^n_{m=0} 2m+1 = (n+1)^2$. Note that $(n+1)^2-n^2=2n+1$. You have made a sequence where adding a successive term makes the sum equal the successive square. What is the $(n+2)^2-n^2$? Certainly you can set a_1 to be a perfect square, then you are guaranteed to have such a sequence that you want. I hope I made enough sense without giving away too much away.

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## 1. What is an arithmetic series perfect square?

An arithmetic series perfect square is a sequence of numbers where each number is the sum of the previous number and a constant difference. In addition, the sequence must also have a perfect square as one of its terms.

## 2. How can I identify if a series is an arithmetic series perfect square?

To identify an arithmetic series perfect square, you can look for a constant difference between each term and check if one of the terms is a perfect square. Another method is to check if the series follows the formula: nth term = a + (n-1)d, where "a" is the first term, "n" is the term number, and "d" is the constant difference.

## 3. What is the formula for finding the sum of an arithmetic series perfect square?

The formula for finding the sum of an arithmetic series perfect square is: Sn = [n/2(2a + (n-1)d)]^2, where "Sn" is the sum of the first "n" terms, "a" is the first term, and "d" is the constant difference.

## 4. Can an arithmetic series perfect square have a negative difference?

Yes, an arithmetic series perfect square can have a negative difference. This means that the series is decreasing instead of increasing. The same formula can be used to find the sum, but the difference will be subtracted instead of added.

## 5. How is an arithmetic series perfect square used in real life?

An arithmetic series perfect square can be used in various real-life scenarios, such as calculating the depreciation of an asset over time, determining the total cost of an installment plan, and predicting population growth. It can also be applied in mathematical and scientific research to analyze patterns and make predictions.

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