Respuesta :
Answer:
The sum of all the even integers between 99 and 301 is 20200
Step-by-step 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 [tex]S_{n}[/tex]= [tex]\frac{n }{2}[ 2a + (n - 1)d][/tex]
Where [tex]n[/tex] is the number of terms
[tex]a[/tex] is the first term
and [tex]d[/tex] 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
[tex]a[/tex] = 100
[tex]d[/tex] can be calculated by taking the difference of the first two terms or any two consecutive terms of the AP
Hence, [tex]d[/tex] = 102 - 100 = 2
[tex]n[/tex] can be determined from the formula used to calculate the Last term of an AP
[tex]l = a + (n - 1)d[/tex]
Where [tex]l[/tex] is the last term
In the AP, the last term, [tex]l[/tex] = 300
Then
[tex]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[/tex]
Hence, there are 101 terms in the AP
Now, to determine the sum
[tex]S_{n}[/tex]= [tex]\frac{n }{2}[ 2a + (n - 1)d][/tex]
[tex]S_{n} =\frac{101}{2}[ 2(100) + (101 - 1)2]\\ S_{n}=\frac{101}{2}[ 200 + (100)2]\\ S_{n}=\frac{101}{2}[ 200 + 200]\\S_{n}=\frac{101}{2}[ 400]\\S_{n} = 20200[/tex]
Hence, the sum of all the even integers between 99 and 301 is 20200
