Competition · AMC preparation · step 4 of 4

AMC 8 · 2018 · #2

Grade 5 arithmeticpattern

Archetype Telescoping Collapse └ Sub-archetype Product Ratio Cancellation

fraction-arithmeticfraction-multiplicationpattern-recognition pattern-recognitionidentify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

Compute the value of the product (1+

\frac{1}{1}

)(1+

\frac{1}{2}

)(1+

\frac{1}{3}

)(1+

\frac{1}{4}

)(1+

\frac{1}{5}

)(1+

\frac{1}{6}

) and match the result to one of the five answer choices.

Pick an answer.

(A)

$\frac{7}{6}$

(B)

$\frac{4}{3}$

(C)

$\frac{7}{2}$

(D)

(E)

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

Each factor 1 + 1/k rewrites cleanly as (k+1)/k, so the product becomes 2/1·3/2·4/3…7/6 — a chain where each numerator matches the next denominator. Tool #5 (Look for a Pattern) is exactly the move that spots this telescoping cancellation, turning a six-fraction multiplication into a one-step answer. Tool #9 (Easier Problem) backs it up: trying the same product with only 2 or 3 factors first reveals the rule "the answer is just the last numerator" before we trust it on all six.

1STEP 1

Rewrite each factor as a fraction

Rewrite each factor 1 + $\frac{1}{k}$ as $\frac{k+1}{k}$ , so the six factors become $\frac{2}{1}$ , $\frac{3}{2}$ , $\frac{4}{3}$ , $\frac{5}{4}$ , $\frac{6}{5}$ , $\frac{7}{6}$ .

1+1/k = (k+1)/k → 2/1, 3/2, 4/3, 5/4, 6/5, 7/6

2STEP 2

Try smaller cases first

Test smaller cases (Tool #9): 2 factors give 3, 3 give 4, 4 give 5 — the pattern leaves only the last numerator.

2/1·3/2 = 3, 2/1·3/2·4/3 = 4, 2/1·3/2·4/3·5/4 = 5

3STEP 3

Cancel across the product

Apply the telescoping cancellation to all six factors: every inner number cancels, leaving $\frac{2}{1}$ · $\frac{3}{2}$ · $\frac{4}{3}$ · $\frac{5}{4}$ · $\frac{6}{5}$ · $\frac{7}{6}$ = $\frac{7}{1}$ .

2/1·3/2·4/3·5/4·6/5·7/6 = 7/1

4STEP 4

Read off the value

Read off the value: $\frac{7}{1}$ = 7, which matches choice (D).

7/1 = 7 → (D)

Answer

Each factor is greater than 1, so the product must be greater than 1 — that already eliminates nothing, but the smallest factor is

\frac{7}{6}

≈ 1.17 and the largest is 2, so a rough estimate 2 × 1.5 × 1.33 × 1.25 × 1.2 × 1.17 ≈ 7 matches the exact answer. The pattern check is even stronger: with n factors of the form 1+

\frac{1}{k}

for k=1… n, the product equals n+1. Here n = 6, so the answer is 7 — choice (D).

💡Key takeaway

This AMC 8 problem only needs Grade 5 fraction multiplication you already know — once you spot the cancellation pattern, the answer falls out in one line!

Rewrite each factor as a fraction
Try smaller cases first
Cancel across the product
Read off the value

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Rewrite each factor as a fraction

Try smaller cases first

Cancel across the product

Read off the value

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