Answer:
[tex]a_n=15n+100[/tex] where [tex]n =1, 2, 3, .....[/tex]
Step-by-step 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 [tex]n^{th}[/tex] term of the given AP.
Formula for [tex]n^{th}[/tex] term:
[tex]a_n=a+(n-1)d[/tex]
[tex]a[/tex] is the first term and
[tex]d[/tex] is the common difference.
Here, [tex]a[/tex] = 115 (because $100 is the fixed amount so first term is taken as 115 when one book is read)
[tex]d = 15[/tex]
Putting the values, we get:
[tex]a_n=115+(n-1)15\\\Rightarrow a_n=115+15n-15\\\Rightarrow a_n=15n+100[/tex]
Therefore, formula to represent the situation is:
[tex]a_n=15n+100[/tex] where [tex]n =1, 2, 3, .....[/tex]
