Depth · one concept from first principles · step 2 of 4
Find the rule in a fraction sequence
The fractions below are listed following a fixed rule. Find the fraction that will be placed in the 41st position.
OperationsFractions
Your answer
Watch out This list keeps fractions unreduced — it shows , not — so don't simplify your final fraction.
How to solve
Strategy Look for a Pattern — The list groups by denominator, and the group with denominator (k+1) contributes k terms. Adding up these group sizes (1+2+3+...) tells me which group position 41 falls in, then I count within that group.
1STEP 1
See the group sizes
Each denominator d holds d-1 fractions, so the running totals are 1, 3, 6, 10, 15, 21, 28, 36, 45.
1, 1+2=3, +3=6, +4=10, +5=15, +6=21, +7=28, +8=36, +9=45
Each new denominator adds one more fraction than the previous group, a clear growing pattern.
3.OA.D.9Look For A Pattern2STEP 2
Locate group for position 41
Denominator 9 ends at position 36. The next group, denominator 10, covers 37-45 — so 41 falls in the denominator-10 group.
Using the running totals reduces a far-off 41st term to 'which short group is it in'.
3.OA.D.9Solve An Easier Related ProblemThe 41st fraction in the list belongs to the group whose denominator is 10.
Why?
The groups for denominators 2 through 9 fill exactly the first 36 positions, and the next group, denominator 10, fills positions 37 through 45, so position 41 lands inside it.
Why?
Each group holds one less fraction than its denominator, so denominators 2 through 9 make groups of size 1, 2, 3, up to 8, which add to 36.
Why?
Inside a group the numerators run 1, 2, 3, up to one less than the denominator, and there is exactly one fraction for each numerator, so counting the numerators counts the fractions.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The groups sit one right after another with no gaps and no overlaps, so adding up the group sizes gives the total positions used, which tells where each group ends.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Find the position within the group
Position 41 is the (41 - 36) = 5th fraction in the denominator-10 group, so its numerator is 5.
Within a group the numerator equals the place number, so the 5th term has numerator 5.
3.NF.A.1Look For A PatternAnswer
41 - 36 = 5 →
Positions 37-45 hold through ; the 41st is the 5th of these, which is . This fits the pattern (numerator counts up, denominator fixed at 10). The numerator 5 is between 1 and 9, as required for that group.
Takeaway
This only needs Grade 3 pattern sense: add up the group sizes until you reach 41, then count inside that group!
- See the group sizes
- Locate group for position 41
- Find the position within the group
Where next?
Another one like thissuggested
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →