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

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 =

Where is the number of terms

is the first term

and 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

= 100

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

Hence, = 102 - 100 = 2

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

Where is the last term

In the AP, the last term, = 300

Then

Hence, there are 101 terms in the AP

Now, to determine the sum

=

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

a is the right answer

step-by-step explanation: so we look at the formula un = a + (n-1)d

un is the amount of seats

n is the nth term aka row number

a is the original sequence/row number

d is the difference between each term

so in this a=22 d=6 and n can remain as n because we are simply formjng a rule

lets put these in the equation

un=22+(n-1)(6)

this is your answer to part a

however

lets double check that this is correct, we'll try putting it into the second term

un=a+(n-1)d

u2=22+(2-1)(6)

u2=22+6

u2=28

as the question said the second row has 28 seats so now we know the formula we made is correct.

as for part b

un=a+(n-1)

its asking us to find a row with 100 seats.

remember un was the amount of seata so all we do is substitute this in.

un=100 a=22 d=6

un=a+(n-1)d

100=22+(n-1)(6)

so now we simplify to find n which is the row number.

100-22=22+(n-1)(6)-22

78=(n-1)(6)

78/6=(n-1)(6)/6

13=n-1

13+1=n-1+1

14=n

so now we know that in the 14th row there is 100 seats.

hope this and that you understood! its my first time using the app or anyone out on here : p its also me to revise so a bunch! ask more questions if you dont understand ☺