Problem 19 · 1988 AJHSME
Hard
Algebra & Patterns
arithmetic-sequence-nth-term
What is the 100th number in the arithmetic sequence: 1, 5, 9, 13, 17, 21, 25, …?
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Answer: A — 397.
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Hint 1 of 2
Each term is 4 more than the one before. The catch: getting from the *1st* term to the *100th* term takes how many steps of 4 — 100, or one fewer?
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Hint 2 of 2
It's 99 steps, not 100, because the first term needs no step to reach itself. So the 100th term = 1 + 99 × 4.
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Approach: start value + (number of steps) × step size
- The list climbs by 4 each time. To reach the 100th term from the 1st you take 99 steps (the 2nd is 1 step out, the 3rd is 2 steps out, … the 100th is 99 steps out). So the 100th term = 1 + 99 × 4 = 1 + 396 = 397.
- Trap to avoid: choice 401 comes from using 100 steps instead of 99. The number of *gaps* between terms is always one less than the number of terms — the same reason a fence with 100 posts has only 99 gaps.
- Why this transfers: any evenly-spaced list works as start + (n − 1) × step. The (n − 1) is the fence-post idea: count the gaps you cross, not the terms you land on.
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