How to calculate the first hundred terms of a sequence​

Question: what is the sum of the first 100 whole numbers?? how am i supposed to work this out efficiently? thanks

The question you asked relates back to a famous mathematician, Gauss. In elementary school in the late 1700’s, Gauss was asked to find the sum of the numbers from 1 to 100. The question was assigned as “busy work” by the teacher, but Gauss found the answer rather quickly by discovering a pattern. His observation was as follows:

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

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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.

Hope this helps!

Explanation:

Is the sequence arithmetic, does it add the same thing every time? If so take the first term and add 99 times the common difference (the number being added each time).

Example: 5, 18, 31, 44, … The common difference is 13. 5 + 99(13) = 1292

Is the sequence geometric, does it multiply by the same thing every time? If so take the first term and multiply by the common ratio (the number multiplied by each time) to the 99th power.

Example 13, 39, 117, 351, … The common ratio is 3. 13 x 3⁹⁹ = 2.333 x 10⁴⁸