Pattern in equations predicts the answer

4.OA.C.53.OA.D.9 · take · grade 4

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

Find the rule in the calculations, then write the seventh calculation.

Position	Calculation
1st	$1$
2nd	$1+3+5=9$
3rd	$1+3+5+7+9=25$
4th	$1+3+5+7+9+11+13=49$
5th	$1+3+5+7+9+11+13+15+17=81$

Show solution

Understand

A list of calculations adds up consecutive odd numbers starting from 1. The 1st calculation is just 1, the 2nd is 1+3+5=9, the 3rd is 1+3+5+7+9=25, and so on, with each calculation using more odd numbers than the one before. We must write the seventh calculation, including its result, without grinding through every sum.

Givens

1st: 1
2nd: 1+3+5 = 9
3rd: 1+3+5+7+9 = 25
4th: 1+3+5+7+9+11+13 = 49
5th: 1+3+5+7+9+11+13+15+17 = 81

Unknowns

The seventh calculation: which odd numbers it adds and what it equals

Constraints

Each calculation adds consecutive odd numbers starting at 1
The number of terms grows by 2 each step: 1, 3, 5, 7, 9, ...

Plan

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

The results 1, 9, 25, 49, 81 are the square numbers 1, 3, 5, 7, 9 squared, and the number of terms grows by 2 each step. Spotting these two patterns lets us write the seventh calculation and its result directly.

Execute

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

Count how many odd numbers each calculation adds: 1st has 1 term, 2nd has 3, 3rd has 5, 4th has 7, 5th has 9. The term count goes up by 2 each time, so the 6th has 11 terms and the 7th has 13 terms.

$1,\ 3,\ 5,\ 7,\ 9,\ 11,\ 13$ terms

Each step just tacks on the next two odd numbers, so the count of terms climbs by 2 every time.

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

The 7th calculation has 13 terms, so it runs through the first 13 odd numbers, ending at the 13th odd number, which is 2 x 13 - 1 = 25.

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25

The odd numbers in order are 1, 3, 5, ...; the 13th of them is 25, so that is where the sum stops.

#9 Solve an Easier Related Problem 3.OA.D.9

Look at the results: 1 = 1x1, 9 = 3x3, 25 = 5x5, 49 = 7x7, 81 = 9x9. The result is always the number of terms multiplied by itself. The 7th calculation has 13 terms, so its result is 13 x 13.

13 \times 13 = 169

Adding the first few odd numbers always builds a perfect square, so we can predict the total without adding all 13.

Answer:

1+3+5+7+9+11+13+15+17+19+21+23+25 = 169

Review

The result must be a square: 169 = 13 x 13, and the calculation uses 13 terms ending at 25, both matching the pattern. It is larger than the 5th calculation's 81, which is right since we added more odd numbers.

Add directly in pairs (tool 9): pair the 13 terms as (1+25)+(3+23)+(5+21)+(7+19)+(9+17)+(11+15) = 26 x 6 = 156, plus the leftover middle term 13, giving 156 + 13 = 169.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Extending the term-count pattern to find the 7th calculation has 13 terms ending at 25
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing the results 1, 9, 25, 49, 81 as squares to predict 13 x 13 = 169

💡 Adding odd numbers starting at 1 always makes a square, so the 7th calculation (13 odd numbers) equals 13 x 13 = 169 with no long adding!