Difference between revisions of "Sums and Perfect Sqares"

Revision as of 12:47, 14 June 2019

Here are many proofs for the Theory that $1+2+3+...+n+1+2+3...+(n-1)=n^2$

PROOF 1: $1+2+3+...+n+1+2+3...+(n-1)=n^2$ , Hence $\frac{n(n+1)}{2}+\frac{n(n+1)}{2}=n^2$ . If you dont get that go to words.Conbine the fractions you get $\frac{n(n+1)+n(n-1)}{2}$ . Then Multiply: $\frac{n^2+n+n^2-n}{2}$ . Finnaly the $n$ 's in the numorator cancel leaving us with $\frac{n^2+n^2}{2}=n^2$ . I think you can finish the proof from there.

PROOF 2: The $1+2+\cdots+n$ part refers to an $n$ by $n$ square cut by its diagonal and includes all the squares on the diagonal. The $1+2+\cdots+ n-1$ part refers to an $n$ by $n$ square cut by its diagonal but doesn't include the squares on the diagonal. Putting these together gives us a $n$ by $n$ square.

PROOF 3: We proceed using induction. If $n = 1$ , then we have $1+0=1^2$ . Now assume that $n$ works. We prove that $n+1$ works. We add a $2n+1$ on both sides, such that the left side becomes $1+2+\cdots + (n+1)+1+2+\cdots + n = n^2 + 2n + 1 = (n+1)^2$ and we are done with the third proof.

PROOF 4: 1 2 3 4 5 ... n 0 1 2 3 4 ... (n-1) ________________ 1 3 5 7 9 ... 2n-1

and the sum of the first $n$ odd numbers is $n^2$ .

Math is like art in many ways and people sometimes make a hobby of proof after proof after proof. However, these proofs are mostly from the Pythagorean Theorem. This theorem already has more proofs then needed so mathematicians soulds make a hobby of making proofs for theorems like these.

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-Proof 2: The <math>1+2+\cdots+n</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal and includes all the squares on the diagonal. The  <math>1+2+\cdots+ n-1</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal but doesn't include the squares on the diagonal. Putting these together gives us a <math>n</math> by <math>n</math> square.
+PROOF 2: The <math>1+2+\cdots+n</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal and includes all the squares on the diagonal. The  <math>1+2+\cdots+ n-1</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal but doesn't include the squares on the diagonal. Putting these together gives us a <math>n</math> by <math>n</math> square.
-Proof 3: We proceed using induction. If <math>n = 1</math>, then we have <math>1+0=1^2</math>. Now assume that <math>n</math> works. We prove that <math>n+1</math> works. We add a <math>2n+1</math> on both sides, such that the left side becomes <math>1+2+\cdots + (n+1)+1+2+\cdots + n = n^2 + 2n + 1 = (n+1)^2</math> and we are done with the third proof.
+PROOF 3: We proceed using induction. If <math>n = 1</math>, then we have <math>1+0=1^2</math>. Now assume that <math>n</math> works. We prove that <math>n+1</math> works. We add a <math>2n+1</math> on both sides, such that the left side becomes <math>1+2+\cdots + (n+1)+1+2+\cdots + n = n^2 + 2n + 1 = (n+1)^2</math> and we are done with the third proof.
+PROOF 4: 1 2 3 4 5 ... n
+1 2 3 4 ... (n-1)
+________________
+3 5 7 9 ... 2n-1
+and the sum of the first <math>n</math> odd numbers is <math>n^2</math>.
+Math is like art in many ways and people sometimes make a hobby of proof after proof after proof. However, these proofs are mostly from the Pythagorean Theorem. This theorem already has more proofs then needed so mathematicians soulds make a hobby of making proofs for theorems like these.

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Difference between revisions of "Sums and Perfect Sqares"

Revision as of 12:47, 14 June 2019