Question

What is the value of 1+2+3+4+5+........96+97+98+99+100?

Answer 1

1 + 2 + 3 + 4 + … + 98 + 99 + 100 Gauss noticed that if he was to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a sum of 101. 1 + 2 + 3 + 4 + 5 + … + 48 + 49 + 50 100 + 99 + 98 + 97 + 96 + … + 53 + 52 + 51 1 + 100 = 101 2 + 99 = 101 3 + 98 = 101 . . . 48 + 53 = 101 49 + 52 = 101 50 + 51 = 101 Gauss realized then that his final total would be 50(101) = 5050. The sequence of numbers (1, 2, 3, … , 100) is arithmetic and when we are looking for the sum of a sequence, we call it a series. Thanks to Gauss, there is a special formula we can use to find the sum of a series: S is the sum of the series and n is the number of terms in the series, in this case, 100.

Answer 2

the following series is in AP.

of the AP -
ist term = 1
common difference = 2-1 = 1
last term = 100
1+(n-1)1 = 100
1+n - 1 = 100
n = 100

so
sum = 100/2 (1+100)
= 50 x 101
=5050