Please help me find the sum

[lostmath]

The directions to this problem are to choose a strategy from the list to solve the following problem: 1) draw a diagram 2) solve a simplier problem 3) work backward 4) guess, check, and revise.

Find the sum of the whole numbers from 1 to 100.

 

[jwpaine]

I'm guessing you mean 1+2+3+4+5....+99+100

This is a famous problem done by an early math student named Carl Friedrich Gauss (born 1777)

He envisioned that if he spread out the numbers 1 through 50 from left to right, and the numbers 51 to 100 from right to left directly below the 1-50 numbers, each combination would add up to 101 (1+100, 2+99, 3+98) etc.

Now....if there are fifty sums, what will your answer be?

Hope this helps.

 

[lostmath]

duh... or in better words I knew there was an easier way to do that than to add 1+2+3+4... thank you very much
lostmath

 

[soroban]

Hello, lostmath!

Here's yet another approach . . . a long one.

Find the sum of the whole numbers from 1 to 100.

We have: \(\displaystyle \:S\:=\:1\,+\,2\,+\,3\,+\,4\,+\,\cdots\,+\,100\)

Consider the partial sums and we get a sequence: \(\displaystyle \:1,\:3,\:6,\:10,\:15,\:21,\:\cdots\)
. . and we want the 100^th term.

These numbers are called Triangular Numbers for obvious reasons.

                                              *
                                 *           * *
                      *         * *         * * *
             *       * *       * * *       * * * *
      *     * *     * * *     * * * *     * * * * *
      1      3        6         10           15

How many are in the 6^th triangular nuimber?
Can we determine the value without counting the dots?

           *
          * *
         * * *
        * * * *
       * * * * *
      * * * * * *

Left-justify the triangle.

      *
      * *
      * * *
      * * * *
      * * * * *
      * * * * * *

Append an inverted copy of the triangle.

      * o o o o o o
      * * o o o o o
      * * * o o o o
      * * * * o o o
      * * * * * o o
      * * * * * * o

We have a \(\displaystyle 6\,\times\,7\) rectangle with 42 dots.

The triangle has half the dots: \(\displaystyle \:\frac{6\,\times\,7}{2}\:=\:21\)


In general, the n^th triangular number is: \(\displaystyle \:T_n\:=\:\frac{n(n\,+\,1)}{2}\)


Therefore: \(\displaystyle 1\,+\,2\,+\,3\,+\,\cdots\,+\,100\;=\;T_{100} \;=\;\frac{100\cdot101}{2}\;=\;5,050\)

 

[Denis]

A bit like jw's:
0+100
1+99
2+98
...
49+51

so:50 *100 = 5000 + the missing 50 = 5050