Easy problems

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May 1st 2006, 07:29 PM
c_323_h

Easy problems

Show that the sum of the first $n$ positive odd integers is $n^2$ .

Show that the sum of the first $n$ positive even integers is $n^2+n$
May 1st 2006, 07:34 PM
ThePerfectHacker

You have,
$1+3+5+...+(2n-1)$
Or,
$(2\cdot 0-1)+(2\cdot 1-1)+...+(2\cdot n -1)$
Regroup,
$2(0+1+2+...+n)-(1+1+...+1)$
Using sum formulas,
$2\cdot \frac{n(n+1)}{2}-n$
Thus,
$n^2+n-n=n^2$
May 1st 2006, 07:51 PM
c_323_h

Quote:

Originally Posted by ThePerfectHacker

You have,
$1+3+5+...+(2n-1)$
Or,
$(2\cdot 0-1)+(2\cdot 1-1)+...+(2\cdot n -1)$

oops, i meant to post this is the calculus section, but i guess it can apply to number theory.

why $(2\cdot 0-1)$ ? This evaluates to -1, and we're looking for the positive integers.
May 9th 2006, 10:37 AM
AfterShock

(2n-1) for definition of odd; start the summation from 1...infinity.

Or, alternatively, 2n+1, and starting it from 0...infinity.
May 9th 2006, 06:55 PM
ThePerfectHacker

You can easily prove these two identities with
Mathematical Induction

I was just too lazy to use induction on these two problems. And decided to use a more elegant way.
May 28th 2006, 07:48 AM
Soroban

Hello, c_323_h!

Quote:

Show that the sum of the first $n$ positive odd integers is $n^2$ .

We have: . $1 + 3 + 5 + \hdots + (2n-1)$

. . an Arithmetic Progression with first term $a = 1$ and common difference $d = 2$ .

The sum of the first $n$ term is an A.P. is: . $S_n\;=\;\frac{n}{2}[2a + d(n-1)]$

So we have: . $S\;=\;\frac{n}{2}[2\cdot1 + 2(n-1)]\;=\;n^2$

Quote:

Show that the sum of the first $n$ positive even integers is $n^2+n$

We have another A.P. . $2 + 4 + 6 + \hdots + 2n$

. . with first term $a = 2$ and common difference $d = 2.$

The sum is: . $S\;=\;\frac{n}{2}[2\cdot2 + 2(n-1)] \;= \;n(n + 1)$