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.
Sahna/Jay -- please use the same name for your posts.
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To determine the number of books in the library's collection after twenty weeks, we need to calculate the total number of books donated during those weeks.
During the first week, ten books were donated.
In each subsequent week, the number of books donated is six more than the previous week.
Let's break this down step-by-step:
1. First, we calculate the number of books donated during the subsequent weeks (weeks 2-20).
- For week 2: 10 + 6 = 16 books donated
- For week 3: 16 + 6 = 22 books donated
- For week 4: 22 + 6 = 28 books donated
- ...
- For week 20: 10 + (20 - 1) * 6 = 10 + 19 * 6 = 10 + 114 = 124 books donated
(Note: We use the formula 10 + (n - 1) * 6, where n represents the number of weeks starting from week 2.)
2. Now we calculate the total number of books donated during the twenty weeks.
- Total books donated = 10 + 16 + 22 + 28 + ... + 124
To find the sum of an arithmetic sequence, we can use the formula: Sn = (n/2) * (a1 + an), where Sn represents the sum of the sequence, n is the number of terms, a1 is the first term, and an is the last term.
In this case, the first term (a1) is 10, the common difference is 6, and the number of terms (n) is 20.
- Total books donated = (20/2) * (10 + 124) = 10 * 134 = 1340
3. Finally, to determine the total number of books in the library's collection after twenty weeks, we add the initial 300 books to the total number of books donated:
- Total books in the library's collection = 300 + 1340 = 1640
Therefore, after twenty weeks, there will be 1640 books in the library's collection.
To determine the number of books in the library's collection after twenty weeks, we need to calculate the number of books donated each week and then accumulate them.
Let's break down the problem week by week:
Week 1: The library starts with 300 books.
Week 2: Ten additional books are donated, bringing the total to 300 + 10 = 310 books.
Week 3: Six more books than the previous week, which means 10 + 6 = 16 books are donated this week. The total collection becomes 310 + 16 = 326 books.
Week 4: Six more books are donated than the previous week, so 16 + 6 = 22 books are donated. The total collection becomes 326 + 22 = 348 books.
We can observe that the number of additional books donated each week is increasing by 6.
Week 5: 22 + 6 = 28 books are donated.
Week 6: 28 + 6 = 34 books are donated.
Week 7: 34 + 6 = 40 books are donated.
As we can see, the number of additional books donated is increasing by 6 each week.
If we continue this pattern, we arrive at the following donations for each subsequent week:
Week 8: 46
Week 9: 52
Week 10: 58
Week 11: 64
Week 12: 70
Week 13: 76
Week 14: 82
Week 15: 88
Week 16: 94
Week 17: 100
Week 18: 106
Week 19: 112
Week 20: 118
Now, we need to find the total number of books in the library's collection after twenty weeks. To do this, we will sum up the number of books donated each week:
Sum = 300 + 10 + (16 + 22 + 28 + ... + 118) = 300 + 10 + (16 + 22 + 28 + ... + 118)
To find the sum of the arithmetic series (16 + 22 + 28 + ... + 118), we first determine the number of terms in the series. We can use the formula:
n = (a + l) / 2
where a is the first term (16) and l is the last term (118).
n = (16 + 118) / 2 = 67
Now, we calculate the sum of the series:
Sum = (n/2) * (a + l)
Sum = (67/2) * (16 + 118)
Sum = 33.5 * 134
Sum = 4489
Finally, we can calculate the total number of books in the library's collection:
Total Books = 300 + 10 + 4489
Total Books = 3799
After twenty weeks, there will be 3799 books in the library's collection.