Find the sum of the first one thousand positive integers. Explain how you arrived at your result. Now explain how to find the sum of the first n positive integers, where n is any positive integer, without adding a long list of positive integers by hand and without using a calculator
the integers form an arithmetic sequence, with
a = 1
d = 1
So, the sum of the 1st 1000 integers is
1000/2 (2+999) = 500500
This is just an illustration of what Gauss figured out as a kid. You can pair up the numbers starting from the ends
1+1000
2+999
3+998
...
500+501
There are 500 pairs, so the sum is 500*1001 = 500500
To find the sum of the first one thousand positive integers, we can use a formula known as the sum of an arithmetic series. The sum of the first n positive integers is given by the formula:
Sum = (n/2)(first term + last term)
For the first one thousand positive integers, the first term is 1 and the last term is 1000. Plugging these values into the formula:
Sum = (1000/2)(1 + 1000)
= 500(1001)
= 500,500
Hence, the sum of the first one thousand positive integers is 500,500.
Now, let's consider how to find the sum of the first n positive integers, where n is any positive integer, without adding a long list of positive integers by hand and without using a calculator.
We can use the same formula for the sum of an arithmetic series mentioned earlier. In this case, the first term is 1 and the last term is n. Plugging these values into the formula:
Sum = (n/2)(1 + n)
This formula allows us to find the sum of the first n positive integers without the need for manually adding all the numbers or using a calculator.