QuestionWhat is the sum of all even integers from 100 to 300? A. 10,100 B. 20,200 C. 22,650 D. 40,200 E. 45,150

Studdy Solution

STEP 1

Assumptions1. The formula for the sum of the first $n$ positive integers is $n(n+1) /$ . . We are looking for the sum of all even integers between99 and301.

STEP 2

First, we need to find the first and last even numbers in the given range. The first even number greater than99 is100 and the last even number less than301 is300.

STEP 3

Next, we need to find the number of even integers in the given range. An even number can be expressed as $2n$ , where $n$ is an integer. So, we can find the number of even numbers by finding the difference between the first and last even numbers, dividing by2 and adding1.
$Number\, of\, even\, integers = (Last\, even\, number - First\, even\, number) /2 +1$

STEP 4

Plug in the values for the first and last even numbers to calculate the number of even integers.
$Number\, of\, even\, integers = (300 -100) /2 +1$

STEP 5

Calculate the number of even integers.
$Number\, of\, even\, integers =200 /2 +1 =101$

STEP 6

Now that we have the number of even integers, we can use the formula for the sum of the first $n$ positive integers to find the sum of these even integers. However, since these are not the first $n$ positive integers, but rather the first $n$ even integers, we need to multiply the result by2.
$Sum\, of\, even\, integers =2 \\times [n(n+1) /2]$

STEP 7

Plug in the value for $n$ to calculate the sum of the even integers.
$Sum\, of\, even\, integers =2 \\times [101(101+1) /2]$

STEP 8

Calculate the sum of the even integers.
$Sum\, of\, even\, integers =2 \\times [101 \\times102 /2] =2 \\times5151 =10302$ The sum of all the even integers between99 and301 is10,302.

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