Problem 9 · 2015 AMC 8
Easy
Algebra & Patterns
arithmetic-sequencearithmetic-series
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working 20 days?
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Answer: D — 400 widgets.
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Hint 1 of 2
Write out the daily sales: 1, 3, 5, 7, … — those are exactly the odd numbers, in order. Over 20 days you're adding the first 20 odd numbers.
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Hint 2 of 2
There's a gem worth memorizing: the sum of the first n odd numbers is always n2 (1 = 12, 1+3 = 22, 1+3+5 = 32, …). So no long addition is needed.
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Approach: recognize the odd numbers; their running total is a perfect square
- Day k sales = 2k − 1, so over 20 days the total is 1 + 3 + 5 + … + 39 — the first 20 odd numbers.
- Key fact: 1 + 3 + 5 + … + (2n−1) = n2. (Picture building an n×n square one L-shaped layer at a time: each new layer adds the next odd number of unit squares.)
- Total = 202 = 400.
Another way — pair the ends (Gauss pairing):
- Pair first-with-last: (1 + 39), (3 + 37), (5 + 35), … Each pair sums to 40.
- The 20 terms make 10 such pairs, so the total is 10 × 40 = 400.
- This is the all-purpose arithmetic-series trick: sum = (number of terms) × (first + last)/2 = 20 · (1 + 39)/2 = 400.
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