Anonymous
Anonymous asked in Science & MathematicsMathematics · 9 years ago

# What is a fast way of adding all the numbers from 1 to 100 ?

i really need to find out the trick ASAP

thans :) xx

Relevance

Imagine a square 4 * 4 dots in size

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You can split it diagonally into two triangles of the same size and a diagonal

The number of dots in the square is 4*4 = 16

The number of dots along the diagonal is 4

The remaining dots are 16 - 4 = 12 and since the triangles are the same size we can find the number by dividing 12 by 2 = 6

Say we want a slightly bigger triangle, one that includes the diagonal then the number of dots would be 6 + 4 = 10

Why do these dots and triangles matter?

The triangle has 1 dot in the first row, 2 in the second, 3 in the third etc. so knowing the number of dots in the triangle here tells us the sum of 1, 2 and 3 = 6

More generally if when had a square with n dots by n dots there would be n² dots in total.

Along the diagonal there would be n dots, so both triangles would have (n² - n)/2

To create the bigger triangle:

(n² - n)/2 + n = (n² - n)/2 + 2n/2 = (n² - n + 2n)/2 = (n² + n)/2

This can be written as n(n+1)/2

In the case of n = 100 rows then there are 100*101/2 = 5050 dots which is also the sum from 1 to 100.

• You can use the sum formula for arithmetic series

n=100 (No. of values)

d=1 (Difference between two consecutive values)

a1=1 (First value of series)

Sum=S= n/2 (2a1 + (n-1)(d))

S=100/2 (2 + 99(1))

S=50(101)

S= 5050

• arithematic progression

tn-1=a+d(n-1) this is the formula

100=1+1(n-1)

100=1+n-1

100=n

n/2(first term+last term)

100/2(1+100)

50(101)=5050

• hmmmm...hmmmm...

all the numbers? ALL the numbers? Every bloomin' number?!!

yeah right.

would you be meaning: whole numbers, natural numbers, rational numbers, integers, squared numbers...???

need just a tad bit more information to provide a constructive answer. based on the information in the question, the 'fast way' would be pencil and paper.

good luck.