i need a shortcut to find the sum of 1-100 numbers?

Answer 1

it is n(n+1)/2 put 100 in n 100*101/2=5050

Answer 2

As mentioned, the sum of the first n numbers is n(n+1)/2. The formula can be proved easily by the following argument: Take your sum 1+2+3+...(n-2)+(n-1)+n. Call it S. Reverse the order of the numbers in it, so you have n+(n-1)+(n-2)+...3+2+1. These sequences have the same sum and the same number of terms (since one is just a permutation of the other). Now add the sequences together term by term - you get (n+1)+(n-1+2)+(n-2+3)+...(3+n-2)+(2+n-1)+(1+n). Note that every term in this new sequence reduces to n+1, and has n terms. So the sum of this new sequence is simply n*(n+1). However this sequence was produced by adding S to itself. Thus 2S=n(n+1) and S=n(n+1)/2

Alternatively, a proof by induction:

Base case: Denote the sum of the first n numbers by ∑n. ∑1=1 and 1*(1+1)/2=1, therefore this formula holds when n=1.

Inductive step: suppose that the formula holds for some n. Note that ∑(n+1)=n+1+∑n. Since the formula holds for n, we can substitute, giving n+1+n(n+1)/2. By simple algebra, we get 2(n+1)/2+n(n+1)/2, giving (2+n)(n+1)/2 which is (n+1)((n+1)+1)/2, which is what we get when substituting n+1 into our formula. Therefore the formula holds for n+1.

Therefore by induction, ∑n=n(n+1)/2 for all n. Q.E.D.

Answer 3

take this simple steps... 1 to 100... there are 100 numbers.. even number.. if its even number, divide the largest even number by 2, which in this case gives 50. add the smallest and largest number... which in this case, gives 101. thus 50 x 101 = 5050. If the total number of numbers is odd, lets say 1-99, there are 99 numbers. 99 / 2 = 49.5 Round down to the nearest whole number, which becomes 49. 49 x (1+99) = 4900... but remember there is 1 more number? add that number in... which in this case, is 50.. so 1+2+3+...+98+99 = 4950

Answer 4

The formula to find out the sum of first n natural numbers is {n(n+1)} / 2

For example sum of first 10 natural numbers is 10(11)/2 = 55
In the similar way you can find it for 100 it is as follows
100(101)/2 = 5050.

Similarly you can do it for any numbers.

Hope you understood it.

Answer 5

100+1 = 101
99+2 = 101
98+3 = 101
.
.
.
51+ 50 = 101

So -- 50*101 = 5050.

Gauss knew this at age 9


In general, the sum of the first N numbers is equal to N*(N+1)/2

Answer 6

Formula is N*(N+1)/2 for sum of intergers from1 to N.

100*101/2 = 5050

Answer 7

you have to use the formula, Sum of n numbers = n(n+1)/2

here n = 100, and hence,
1+2+3+......+100 = 100(99)/2 = 50 x 99 = 4950

Answer 8

(100/2) * 101 = 50 * 101 = 5050

Here is how it is done

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
4 + 97 = 101
...

Answer 9

Here's a hint ...

Can you think of a way to pair up the numbers so that each pair has the same sum? How many pairs are there?

Actually there is a famous story about a famous mathematician, Gauss, who got this problem in school and figured out this trick...

Answer 10

Spreadsheet program. Drag to highlight the numbers, press the summation button and voila!!

The "manual" method is to count the identical values, and multiply the value against the count. Then add the multiplied values.