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.
300+10+16+22+...
= 310 + 19/2 (2*16 + 18*6)
= 1640
To find out how many books are in the library's collection after twenty weeks, we need to calculate the number of books donated each week and then add them all up.
Since the library initially has 300 books, we can start by adding the ten books donated in the first week. This brings the total to 300 + 10 = 310 books.
In the second week, six more books are donated than in the previous week. So, we add 10 + 6 = 16 books to the collection, bringing the total to 310 + 16 = 326 books.
Similarly, in the third week, we add 16 + 6 = 22 books to the collection, bringing the total to 326 + 22 = 348 books.
We continue this pattern for each subsequent week, adding six more books than the previous week's donation:
Week 4: 22 + 6 = 28 books added (total: 348 + 28 = 376 books)
Week 5: 28 + 6 = 34 books added (total: 376 + 34 = 410 books)
...
Week 20: ???
To calculate the total number of books after twenty weeks, we need to add up the donations for each week:
Week 1: 10 books
Week 2: 16 books
Week 3: 22 books
...
Week 20: ??? (this is what we want to find)
To calculate the number of books donated in week 20, we can use a mathematical formula for arithmetic series:
nth term = first term + (n - 1) * common difference
Here, n is the number of weeks (20 in this case), the first term is 10, and the common difference is 6.
Using the formula:
20th term = 10 + (20 - 1) * 6 = 10 + 19 * 6 = 10 + 114 = 124 books
Therefore, the library's collection will have 300 + 10 + 16 + 22 + ... + 124 books after twenty weeks. We can calculate this sum using the formula for the sum of an arithmetic series:
Sum = [n/2] * (first term + last term)
Here, n is the number of terms (20 in this case), the first term is 10, and the last term is 124.
Using the formula:
Sum = [20/2] * (10 + 124) = 10 * 134 = 1340 books
After twenty weeks, the library's collection will have a total of 1340 books.