Depth · one concept from first principles · step 2 of 4
Pattern in equations predicts the answer
Find the rule in the calculations, then write the seventh calculation.
| Position | Calculation |
|---|---|
| 1st | |
| 2nd | |
| 3rd | |
| 4th | |
| 5th |
Operations
Your answer
Watch out It's tempting to match the '7th' calculation with a calculation that has 7 terms (giving 49), but the term count grows by 2 each step, so the 7th calculation actually has 13 terms.
How to solve
Strategy Look for a Pattern — 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.
1STEP 1
Count the terms in each calculation
Term counts go 1, 3, 5, 7, 9, 11, up by 2 each time, so the 7th calculation 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.
4.OA.C.5Look For A Pattern2STEP 2
Find the last odd number used
With 13 terms it stops at the 13th odd number, 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.
4.OA.C.5Look For A Pattern3STEP 3
Use the square-number pattern for the result
Results 1, 9, 25, 49, 81 are each the number of terms multiplied by itself, so 13 terms gives 13 x 13.
13 × 13 = 169
Adding the first few odd numbers always builds a perfect square, so we can predict the total without adding all 13.
3.OA.D.9Solve An Easier Related ProblemThe seventh calculation's total equals the number of odd numbers it adds, 13, multiplied by itself, because adding consecutive odd numbers starting at 1 always builds a perfect square.
Why?
Show the running total as dots packed into a square, and each odd number you add becomes the L-shaped strip that grows the square to the next size up.
Why?
To grow a square to a side one dot longer, you lay down a new bottom row and a new side column of the same length and let them share the single corner dot, so the strip always holds an odd number of dots.
Why?
The larger square is exactly the smaller square plus that corner strip, with no dot left out and none counted twice, so the strip is just the extra dots needed to complete it.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Once the last odd strip is in place the dots fill a complete square, with as many rows as there are dots in each row, and that many rows of that many dots is that number times itself.
Why?
Counting equal rows of dots is the same as counting that many groups of that size, which is exactly what multiplying the two counts means.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
13 × 13 = 169
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.
Takeaway
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!
- Count the terms in each calculation
- Find the last odd number used
- Use the square-number pattern for the result
Where next?
Another one like thissuggested
Step back and solidifysuggested
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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 →