SOLUTION: If you add consecutive odd integers beginning with 1, the sum will always be a square. Why is this true? How do I prove this algebraically?

Question 413850: If you add consecutive odd integers beginning with 1, the sum will always be a square. Why is this true? How do I prove this algebraically?
Answer by bayners123(12) (Show Source): You can put this solution on YOUR website!
If you write out a sequence with even intervals between the values of the numbers, then to find the sum of the sequence you can find the average and multiply by the number of terms you have. As the numbers are evenly placed, to find the average you can just add the last and first and then divide by 2.
In this case that means you have the first term, 1, and the last term which will be 2n-1 for a series with n numbers in it. This makes the average
which is just n.
Now to find the sum, just multiply the average, n, by the number of terms, also n, to find the sum of the series is, as anticipated,
RELATED QUESTIONS
These are my homework questions that I am confused on. Quetion 1 If x is any real... (answered by Fombitz,MathLover1)

Would you pls help me to answer this: 3. Arianna is trying to determine if the following (answered by jim_thompson5910)

This is a true/false question from my test. If two pairs of consecutive sides of a... (answered by scott8148)

How do I prove that a prime number cannot be a sum of two or more consecutive positive... (answered by solver91311)

If x is the sum of seven consecutive odd integers beginning with 3 and y is the sum of... (answered by checkley77)

is this always, sometimes or never true? the sum of 3 consecutive odd integers equals an... (answered by josmiceli)

Prove the following conjecture: "The sum of any three positive consecutive odd integers... (answered by Alan3354)

If the sum of n consecutive integers is 0, which of the following must be true? 1) n... (answered by scott8148)