Question

Gregory has agreed to donate $100 to Spring Valley High School for its library. In addition, he will donate $15 for every book a student at Spring Valley High School reads during the summer. The sequence shown represents the possible amounts that Gregory will be donating for the summer. 100, 115, 130, 145, 160,…. Write explicit formula represents this problem situation?

Answer 1

`a_n=15n+100` where
`n =1, 2, 3, .....`

Explanation:

Given that:

Amount of money to be donated by Gregory = $100

Amount of money to be donated for every book that a student will read = $15

The series becomes:

100, 115, 130, 145, 160, .....

To find:

The explicit formula to represent this problem situation.

Solution:

First of all, let us have a look at the difference of each term from its previous term.

Second term - First term = 115 - 100 = 15

Third term - Second term = 130 - 115 = 15

Fourth term - Third term = 145 - 130 = 15

The difference is same, therefore the given series is an Arithmetic Progression (AP).

We have to find the
`n^(th)` term of the given AP.

Formula for
`n^(th)` term:

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

`a` is the first term and

`d` is the common difference.

Here,
`a` = 115 (because $100 is the fixed amount so first term is taken as 115 when one book is read)

`d = 15`

Putting the values, we get:

`a_n=115+(n-1)15\\\Rightarrow a_n=115+15n-15\\\Rightarrow a_n=15n+100`

Therefore, formula to represent the situation is:

`a_n=15n+100` where
`n =1, 2, 3, .....`