Can you explain this please $T(n) = (n-1)+(n-2)+…1= \frac{(n-1)n}{2}$

Question

Possible Duplicate:
Proof for formula for sum of sequence 1+2+3+…+n?

Can you explain this please $$T(n) = (n-1)+(n-2)+…1= \frac{(n-1)n}{2}$$

I am really bad at maths but need to understand this for software engineering.

Answer 1

For example, if we wish to add the numbers

$$ 1+2+3+4+5+6 $$

you can pair them as such

$$ (1+6)+(2+5)+(3+4)=7+7+7=21. $$

But, here $n=7$ and there are $\frac{n-1}{2}=3$ pairs of $n=7$. Thus we have that the sum is $\frac{n-1}{2}\cdot n=3\cdot 7=21$. In general, you can pair the sum

$$ 1+2+3+\cdots +(n-2)+(n-1) $$

as pairs $(n-1+1), (n-2+2), (n-3+3), \ldots$ and so forth. This will give you $\frac{n-1}{2}$ pairs of $n$ if $n-1$ is even. If $n-1$ is odd, leave off $n-1$ and do it for $1+\cdots +(n-2)$ and add on $n-1$ to obtain the same formula.

Answer 2

It says that the following two programs give the same output:

int mySum( int integer)
{
if( integer == 1)
     return 1;
else
       {
       return(integer+(mySum(integer-1));
       }
}

and

int mySum2( int integer)
{
   return(integer*(integer+1)/2);
}

Answer 3

Here is a story that is attributed to Gauss.

His teacher asked him to sum $1+2+3+...+99$. Then Gauss summed this terms with the same terms but with the reverse order, i.e., $(1+99)+(2+98)+(3+97)+...+(98+2)+(99+1).$ The trick here is that the sum of each term in paranthesis is 100 and we have 99 terms. But he summed the terms twice, so he needed to divide the sum by 2. Hence the sum is $(99\times 100) /2.$

So if we generalize this, we get the formula.

Answer 4

Edited to correct the attribution to Gauss as noted by @KaratugOzanBircan

By Gauss:

$\begin{array}{ccccc}T(n)&=&(n-1)&+&(n-2)&+&\cdots&+&2&+1\\T(n)&=&1&+&2&+&\cdots&+&(n-2)&+&(n-1)\end{array}$

---

$2T(n)=n+n+\cdots+n$ ($n-1$ times)=$n.(n-1)$

Hence $T(n)=\frac{n(n-1)}{2}$

Answer 5

Say, $$K = (n-1) + (n-2) + \cdots + 2 + 1$$

Writting it backwards: $$K = 1 + 2 + \cdots + (n-2)+(n-1)$$

Lets add the two equations, which gives, $$2k = n + n + \cdots + n$$ Note that each term is $n$ and there are exactly $(n-1$) terms, so$$2k = n + n + \cdots + n = n(n-1)$$

Dividing by $2$: $$K= (n-1) + (n-2) + \cdots + 2 + 1 = \frac{n(n-1)}{2}$$

For more of intutive proofs of this identity see here.