Question

For any positive integer n, the sum of the first n positive integers equals n ( n 1 ) 2 . What is the sum of all the even integers between 99 and 301

Answer 1

The sum of all the even integers between 99 and 301 is 20200

Explanation:

For the sum of all the even integers between 99 and 301, this can be calculated using the formula for Sum of n terms in an Arithmetic Progression (AP)

Sum of n terms in an AP
`S_(n)`=
`(n )/(2)[ 2a + (n - 1)d]`

Where
`n` is the number of terms

`a` is the first term

and
`d` is the common difference

The AP for the sum of all the even integers between 99 and 301 will look like

100 + 102 + 104 + ... + 300

`a` = 100

`d` can be calculated by taking the difference of the first two terms or any two consecutive terms of the AP

Hence,
`d` = 102 - 100 = 2

`n` can be determined from the formula used to calculate the Last term of an AP

`l = a + (n - 1)d`

Where
`l` is the last term

In the AP, the last term,
`l` = 300

Then

`300 = 100 + (n - 1)2\\300 - 100 = (n - 1)2\\200 = (n - 1) 2\\(n - 1) = 200 / 2\\n - 1 = 100\\n = 100 + 1\\n = 101`

Hence, there are 101 terms in the AP

Now, to determine the sum

`S_(n)`=
`(n )/(2)[ 2a + (n - 1)d]`

`S_(n) =(101)/(2)[ 2(100) + (101 - 1)2]\\ S_(n)=(101)/(2)[ 200 + (100)2]\\ S_(n)=(101)/(2)[ 200 + 200]\\S_(n)=(101)/(2)[ 400]\\S_(n) = 20200`

Hence, the sum of all the even integers between 99 and 301 is 20200