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A vendor earns $100 in the first week of June and each following week earns $12.50 more than the week before. How much has he earned in total after 12 weeks?
A$2,025.00
B$1,200.00
C$2,850.00
D$2,362.50
Answer & Solution
Correct answer: A. $2,025.00
1. Weekly earnings form an arithmetic sequence with a_1 = $100 and common difference d = $12.50.
2. Find the twelfth week's pay: a_12 = 100 + (12 - 1)(12.50).
3. a_12 = 100 + 137.50 = $237.50 in week 12.
4. Total earnings are the series S_12 = 12(a_1 + a_12)/2.
5. S_12 = 12(100 + 237.50)/2 = 6 x 337.50.
6. 6 x 337.50 = $2,025.00 earned over the 12 weeks.
7. Option B, $1,200, pretends every week paid the starting $100 and ignores the raises entirely.
8. Option C multiplies the final week's $237.50 by 12, and option D averages incorrectly by dropping the first week.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.4 SERIES AND THEIR NOTATIONS_
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