Competition · AMC preparation · step 4 of 4
AMC 10 · 2022B · #9
Grade 7 arithmeticPick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
2021 terms is too many to add directly. Tool #9 (Easier Problem) — start with the first 2 or 3 terms, see what the partial sum looks like. Tool #5 (Pattern) — guess the closed form from the small cases and check on one more. Tool #7 (Subproblems) — break the work into 'rewrite one term cleanly' (so neighbours cancel) and 'sum the rewritten terms'. Once the term n/((n+1)!) is rewritten as a difference 1/n! - 1/((n+1)!), the whole sum telescopes — almost everything cancels.
Compute the first partial sums
Compute the first three partial sums: S₁ = , S₂ = , S₃ = .
Three concrete numerical values give us something to look at — pattern-spotting needs data.
5.NF.A.1Solve An Easier Related ProblemLook for the pattern
Spot the pattern: each Sₙ falls short of 1 by one factorial reciprocal, so conjecture Sₙ = 1 - .
Each partial sum lands a tiny bit short of 1, and the 'shortage' is exactly the next factorial reciprocal.
5.OA.B.3Look For A PatternSplit each term
Split n = (n+1) - 1 to rewrite each term: = - .
Splitting n as (n+1) - 1 turns each term into the difference of two consecutive factorial reciprocals — exactly what makes a chain cancel.
Splitting each numerator into the next number minus one turns every term into a difference of two neighbours.
▸ Why?
That split lets the factorial below cancel against part of the numerator in each piece.
▸ Why?
A quantity over itself is one, and multiplying by one changes nothing, so the pieces simplify cleanly.
Telescope the sum
The differences telescope — inner terms cancel in pairs, leaving S = 1 - .
Every inner term shows up once with a + and once with a - — they cancel like dominoes.
7.EE.A.1Look For A PatternRead off a and b
Match 1 - to a - : a = 1, b = 2022, both positive integers.
Two expressions in the same form match term-by-term.
6.EE.A.4Identify SubproblemsAdd a and b
Add: a + b = 1 + 2022 = 2023, choice (D).
Simple addition reads off the final answer.
4.NBT.B.4Solve An Easier Related ProblemThis AMC 10 problem only needs Grade 7 rewriting an expression you already know — split into - , watch the chain cancel down to 1 - , so a + b = 1 + 2022 = 2023.
- Compute the first partial sums
- Look for the pattern
- Split each term
- Telescope the sum
- Read off a and b
- Add a and b
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