A library opens with a collection of 300 books. During the first week after

the opening, an additional ten books are donated to the library’s
collection. In each subsequent week, six more books are donated to the
library’s collection than were donated in the previous week. Work out how
many books are in the library’s collection after twenty weeks.

310, 316, 328, 346, 370....
I have the answer is 1640 just not sure how to write it as an nth term?

The amount donated starts at 10, then grows by 6 each week. So, the donated amounts are

10,16,22,28,...
So, the donated amount for the nth week is an arithmetic progression with
a = 10, d=6

The total donations in 20 weeks is thus

S20 = 20/2 (2*10 + 19*6) = 1340

Add that to the initial 300 books, and you have 1640, as you said.

The book total in week n is

300 + 10 + (n-1)*6 = 304+6n

To find the nth term of the sequence, you need to determine the pattern in the number of books donated each week.

Let's analyze the pattern:

Week 1: 10 books are donated.
Week 2: 6 more books than the previous week's donations, so 10 + 6 = 16 books are donated.
Week 3: 6 more books than the previous week's donations, so 16 + 6 = 22 books are donated.
Week 4: 6 more books than the previous week's donations, so 22 + 6 = 28 books are donated.
Week 5: 6 more books than the previous week's donations, so 28 + 6 = 34 books are donated.

From the pattern, we can see that the number of books donated each week is increasing by 6 each time.

Now, let's write the sequence in terms of the nth term:

Week 1 (n=1): 10
Week 2 (n=2): 10 + 6 × (n-1) = 10 + 6 × (2-1) = 16
Week 3 (n=3): 10 + 6 × (n-1) = 10 + 6 × (3-1) = 22

Generalizing this pattern, we get the nth term formula:

nth term = 10 + 6 × (n-1)

So, to find the number of books in the library after 20 weeks, we substitute n=20 into the nth term formula:

nth term = 10 + 6 × (20-1)
= 10 + 6 × 19
= 10 + 114
= 124

Therefore, after twenty weeks, the library's collection will have 1240 books.