AMC 10 · 2002 · #19

Grade 6 arithmetic

Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery

sequences-arithmeticsystems-of-equations convert-to-algebra ↑ Prerequisites: sequences-arithmetic

📏 Medium solution 💡 2 insights

Problem

An arithmetic sequence

a_1, a_2, a_3, \ldots

is one in which neighboring terms always differ by the same fixed step. The first 100 terms add up to 100, and the next 100 terms,

a_{101}

through

a_{200}

, add up to 200. Find the size of one step,

a_2 - a_1

Pick an answer.

(A)

0.0001

(B)

0.001

(C)

0.01

(D)

0.1

(E)

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Organize Information in More Ways

The obvious route — find a₁ and the step d separately from the two sum equations — is more work than needed. Tool #15 (Organize Information in More Ways) reorganizes the two totals: instead of adding each block up, subtract the second total from the first block's total term by term. Tool #4 (Introduce a Variable) names the step d = a₂ - a₁, which is exactly the unknown, and writes any term 100 places ahead as a_k+100 = a_k + 100d. Tool #5 (Look for a Pattern) is the payoff: every matched pair in the subtraction differs by the same 100d, so the messy difference of sums collapses into a single count times d, leaving one clean equation to solve.

1STEP 1

Name the common step d

Call the common step $d=a_2-a_1$ . Repeating that step 100 times gives $a_{k+100}=a_k+100d$ for every position $k$ .

d = a₂ - a₁, a_k+100 = a_k + 100d

2STEP 2

Subtract the two sums term by term

Subtract the blocks partner by partner: all 100 pairs differ by $100d$ , so $10000d=200-100=100$ .

Σ_k=1¹⁰⁰(a_k+100 - a_k) = 100 × 100d = 10000d = 200 - 100 = 100

3STEP 3

Solve for the step

Divide by 10000: $d=\frac{100}{10000}=\frac{1}{100}$ , so the step $a_2-a_1$ is 0.01 — choice (C).

10000d = 100 → d = 100/10000 = 0.01 = a₂ - a₁ (C)

Answer

0.01

Check the scale. The two blocks differ by only 100 (that is 200 - 100), yet that gap is built from 10000 tiny copies of the step, so the step must be a small fraction — about

\frac{100}{10000}

= 0.01. This rules out the big choices (D) 0.1 and (E) 1, which would make the blocks differ by 1000 or 10000, not 100. It also rules out (A) 0.0001 and (B) 0.001, which would leave a gap of only 1 or 10. Only (C) 0.01 produces the required gap of exactly 100.

💡Key takeaway

When two equal-length blocks of an arithmetic sequence sit a fixed distance apart, subtract them partner by partner — the unknown starting values cancel and only the repeated step is left.

Name the common step d
Subtract the two sums term by term
Solve for the step

Name the common step d

Subtract the two sums term by term

Solve for the step

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