Two Reading-Challenge Rules and the Expression That Misses Week 1
A school library runs a reading-incentive chart. Room 12 reads 14 books as a class in week 1, then reads 9 more books each week than in the week before. Room 13 reads 8 books in week 1 and doubles its weekly total each week after that. The librarian posts both sequences and asks which room will be ahead in week 4. A volunteer writes Room 12's rule for week n as 14 + 9n. Another student objects that substituting n = 1 gives 23, not 14. For Room 13 the volunteer writes "multiply the week before by 2" and claims the room will never catch up because it starts 6 books behind. The librarian wants the class to repair the expression for Room 12, produce a single expression using parentheses for Room 12's total across the first 4 weeks, and explain why a comparison made in week 2 does not predict the result in week 6.
1. Repair the expression for Room 12 so that it gives the correct number of books for week n, verify it for n = 1 and n = 4, and use it to find Room 12's week-4 total.
Read the worked answer
The class adds 9 books after week 1, so week n has had only n − 1 increases of 9:
14 + 9(n − 1)
Checks: n = 1 → 14 + 9(0) = 14; n = 4 → 14 + 9(3) = 41.
The sequence is 14, 23, 32, 41 - a constant difference of 9, which matches the rule.
The volunteer's 14 + 9n adds nine books to week 1 itself, which is why it is 9 too high for every week. That kind of off-by-one error is exactly what a quick substitution test at n = 1 catches before any other work is done.
2. Room 13's totals are 8, 16, 32, 64 for weeks 1–4. Compare the two rooms in week 2 and in week 6, and explain why the week-2 comparison misleads.
Read the worked answer
Room 13 week 6: continue doubling - 128, then 256. Room 12 week 6: 14 + 9(5) = 59.
- Week 2: Room 12 = 23, Room 13 = 16, so Room 12 leads by 7.
- Week 4: Room 12 = 41, Room 13 = 64, so Room 13 leads by 23.
- Week 6: Room 12 = 59, Room 13 = 256.
A week-2 comparison only shows the start. Room 12 grows by the same amount each week (linear, constant difference 9), while Room 13 grows by the same factor each week (doubling, constant ratio 2). A doubling quantity eventually overtakes any constant-difference quantity, no matter how far behind it starts, because each week its gain is a fraction of a total that keeps getting larger. The 6-book head start is quickly irrelevant.
3. Write one expression, using parentheses, for the total number of books Room 12 reads across weeks 1–4, and evaluate it in two different but equivalent ways.
Read the worked answer
Method 1 (add the weekly amounts): 14 + (14 + 9) + (14 + 18) + (14 + 27) = 14 + 23 + 32 + 41 = 110.
Method 2 (group the base amount and the increases): (4 × 14) + 9 × (0 + 1 + 2 + 3) = 56 + 9 × 6 = 56 + 54 = 110.
Parentheses matter here. Writing 4 × 14 + 9 × 0 + 1 + 2 + 3 without grouping would add the increase of 9 to only the first week and then add 1 + 2 + 3 as plain books, giving 56 + 0 + 1 + 2 + 3 = 62, which is far too small.
Both correct methods give 110 because they count the same objects: four copies of the week-1 total plus the extra books added in weeks 2, 3, and 4 (9, 18, and 27, totaling 54).