Use neighbor differences to find rule

4.OA.C.5 · take · grade 4

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

Find the pattern in the number arrangement below, then find the first number in the eighth row.

Row	Numbers
Row 1	$1$
Row 2	$3\ \ 5\ \ 7$
Row 3	$9\ \ 11\ \ 13\ \ 15\ \ 17$
Row 4	$19\ \ 21\ \ 23\ \ 25\ \ 27\ \ 29\ \ 31$

Each row lists consecutive odd numbers: Row 1 has $1$ number, Row 2 has $3$ , Row 3 has $5$ , and Row 4 has $7$ . The count of numbers in each row increases by $2$ from one row to the next.

Show solution

Understand

Consecutive odd numbers are written in rows. Row 1 holds 1 number, Row 2 holds 3 numbers, Row 3 holds 5, Row 4 holds 7, and so on, with each row holding 2 more numbers than the row above it. We must find the first number in the eighth row.

Givens

Row 1: 1
Row 2: 3, 5, 7
Row 3: 9, 11, 13, 15, 17
Row 4: 19, 21, 23, 25, 27, 29, 31
Every entry is an odd number, listed in order with no odd numbers skipped
The count of numbers per row goes 1, 3, 5, 7, ... (increases by 2 each row)

Unknowns

The first number in the eighth row

Constraints

Numbers are consecutive odd numbers (1, 3, 5, 7, ...) flowing row to row
Row n contains 2n - 1 numbers

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

List the first number of each early row (1, 3, 9, 19) and the gaps between them. The gaps grow by a steady amount, which lets us extend the rule down to the eighth row without writing out every odd number.

Execute

#9 Solve an Easier Related Problem 4.OA.C.5

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.

1,\ 3,\ 9,\ 19,\ \ldots

Just looking at the first number of each row turns a big table into a short, friendly list.

#5 Look for a Pattern 3.OA.D.9

Find the jump from one row's start to the next: 3 - 1 = 2, 9 - 3 = 6, 19 - 9 = 10. The jumps are 2, 6, 10, which themselves go up by 4 each time. So the next jumps are 14, 18, 22, 26.

$2,\ 6,\ 10,\ 14,\ 18,\ 22,\ 26$ (each is 4 more than the last)

The jump grows by 4 because each new row adds 2 more odd numbers, and skipping 2 odd numbers means moving ahead by 4.

#5 Look for a Pattern 4.OA.C.5

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.

1 + 2 + 6 + 10 + 14 + 18 + 22 + 26 = 99

Each jump carries us into the next row, so adding the first seven jumps lands us on the start of Row 8.

Answer: 99

Review

Count check: Rows 1-7 hold 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 odd numbers, so Row 8 begins with the 50th odd number, which is 2 x 50 - 1 = 99. This matches, and 99 is odd as required.

Use the count rule directly (tool 9): the numbers before Row n total (n-1) x (n-1) odd numbers, so Row 8 starts at odd number (7 x 7) + 1 = 50th, equal to 99.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the row-start sequence 1, 3, 9, 19 down to the eighth row
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Spotting that the jumps between row-starts grow by 4 each time

💡 Track just the first number of each row and how far it jumps each time, then add the jumps: the eighth row starts at 99!