Depth · one concept from first principles · step 2 of 4
Find the pattern in the number arrangement below, then find the first number in the eighth row.
| Row | Numbers |
|---|---|
| Row 1 | |
| Row 2 | |
| Row 3 | |
| Row 4 |
Each row lists consecutive odd numbers: Row 1 has number, Row 2 has , Row 3 has , and Row 4 has . The count of numbers in each row increases by from one row to the next.
Read off the first entry of each given row: Row 1 starts at 1, Row 2 at 3, Row 3 at 9, Row 4 at 19.
Just looking at the first number of each row turns a big table into a short, friendly list.
4.OA.C.5Solve An Easier Related ProblemThe jumps 3-1=2, 9-3=6, 19-9=10 are 2, 6, 10, each growing by 4, so the next jumps are 14, 18, 22, 26.
The jump grows by 4 because each new row adds 2 more odd numbers, and skipping 2 odd numbers means moving ahead by 4.
3.OA.D.9Look For A PatternThe jump from one row's first number to the next row's first number grows by 4 each time, giving 2, 6, 10, 14, and so on.
Going down a row, the row you leave holds 2 more numbers than the row before it, so this jump crosses 2 more of the little gaps than the last jump did, and each little gap between neighboring numbers is worth 2.
A jump is the whole distance from a row's first number to the next row's first number, and that distance is exactly the little gaps between the numbers you step past, all added together with nothing missed and nothing counted twice.
Two more gaps, each the same size 2, is two equal groups of 2, and adding equal groups is the same as multiplying, so two groups of 2 add 4 to the jump.
Start at Row 1's first number, 1, then add the jumps that lead into Rows 2 through 8: 2, 6, 10, 14, 18, 22, 26.
Each jump carries us into the next row, so adding the first seven jumps lands us on the start of Row 8.
4.OA.C.5Look For A PatternTrack just the first number of each row and how far it jumps each time, then add the jumps: the eighth row starts at 99!
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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 →