Quick calculation:
\[ 1 + 2 + 3 + \ldots + 100 = ? \]
Answer (3)
I read this in a book called Number Games by Ivan Moscovich, "A famous German Mathematician, Karl Fredrich Gauss...When Gauss was a ten-year old student, his teacher wanted to keep his class busy for a long time so, he asked the students to add up all the number from 1 to 100....He realized that the first and last numbers (1 and 100) add up to 101. The second number (2) and the next-to-the-last number also add ed to 101... and so on...He then had 100 pairs, which he multiplied by 101. Since Gauss only wanted the sum of one set of numbers from 1 to 100, he then divided the total in half."
the answer is 5050
Sum of an Arithmetic Series
The sum of numbers from 1 to 100 can be calculated quickly using a formula for the sum of an arithmetic series. We can think of 1+2+3+...+100 as an arithmetic series where each number is one more than the previous number, which means it has a common difference of 1. This series has 100 terms, and we can use the formula for the sum of an arithmetic series, which is:
S = n/2 * (a1 + an)
Where S is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term. So for this series:
n = 100 (since we're adding the first 100 numbers)
a1 = 1 (the first number in the series)
an = 100 (the last number in the series)
Substituting these values into the formula gives us:
S = 100/2 * (1 + 100) = 50 * 101 = 5050
So the sum of numbers from 1 to 100 is 5050.
Understanding the formula for the sum of an arithmetic series allows us to perform calculations swiftly, without needing to add each number individually. This is a life-long skill that highlights the importance of using mathematical shortcuts for efficient problem-solving.
The sum of the numbers from 1 to 100 is 5050. This can be calculated using the formula for the sum of an arithmetic series, which Gauss famously discovered. By pairing numbers, we see that there are 50 pairs of 101, leading to the answer 5050.
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