1+2+3+...+100
how to find the sum of it.
Hi, first thing you do is to want to find Formulas formulas formulas. hey, this is a arithmetic sequence!
An arithmetic sequence is A series such as 3 + 7 + 11 + 15 + ··· + 99 or 10 + 20 + 30 + ··· + 1000 which has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n. The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
The sum of an arithmetic sequence is a_1+a_n/2 *n
All you need to know is that the sum is the first term plus the last term divided by 2 times the number of terms is the sum.
so...1+100 = 101. 101/2 * number of terms is the answer.
one more thing...the way to find how many numbers in a sequence which goes up by increments of 1 is minus the first from last and +1.
thus we have 100-1 = 99. 99+1= 100
so 101 /2 * 100 = 50 * 101 = 5050
There is the old story about the famous mathematician Gauss being given this question along with the rest of the class as "busy work" back in grade school
He thought about it for a while and came up with the answer of 5050
He figured:
the first plus the last = 101
the 2nd plus the 2nd last = 101
the 3rd plus the 3rd last = 101
hey,,,, aren't there just 50 of those sums ?
and he was able to do 50 x 101 = 5050 in his head
To find the sum of the numbers from 1 to 100, you can use a formula or a mathematical technique called arithmetic progression.
Formula method:
The sum of an arithmetic progression is given by the formula:
Sn = n/2 * (a + l)
where Sn is the sum of the arithmetic progression, n is the number of terms, a is the first term, and l is the last term.
In the case of the numbers from 1 to 100, the first term (a) is 1, the last term (l) is 100, and the number of terms (n) is 100.
Using the formula, the sum of the numbers from 1 to 100 can be calculated as:
S100 = 100/2 * (1 + 100) = 50 * 101 = 5050
Arithmetic progression method:
An arithmetic progression has a constant difference between consecutive terms. In the case of the numbers from 1 to 100, the common difference is 1.
To find the sum, you can use the following steps:
1. Add the first and last terms: 1 + 100 = 101.
2. Multiply the sum by half the number of terms: 101 * (100/2) = 101 * 50 = 5050.
Both methods give the same result, which is 5050.