KA-0156Growth patterns and figurate numbers
4 points, difficulty 2 of 3Level 7–8about 105sA pattern is built in rows. The first figure has 1 row, the second has 2 rows, and so on. In every figure, the first row has 1 dot, the second has 3 dots, the third has 5 dots, and each row after has 2 more dots than the one above it. How many dots are in the eighth figure?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Write out the rows for a small figure and add them, before trying figure eight.
2The strategy
Figure 1 has 1 dot. Figure 2 has 1 + 3 = 4. Figure 3 has 1 + 3 + 5 = 9. Those are square numbers.
3The full solution
The nth figure holds n squared dots, so the eighth holds 8 x 8 = 64.
Solution
The reliable way
Build the small cases first. Figure 1 is 1 dot; figure 2 is 1 + 3 = 4; figure 3 is 1 + 3 + 5 = 9; figure 4 is 1 + 3 + 5 + 7 = 16. Those are 1, 4, 9, 16 - the square numbers - so the nth figure holds n squared dots and the eighth holds 8 x 8 = 64. You can also just add the eight odd numbers directly: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64. The transferable idea: build the small cases and read the pattern off them, rather than guessing a formula.
The elegant way
The first n odd numbers always sum to n squared, because they stack into an n by n square.
Why this is on the test: Growth patterns are the cheapest place to practise trying small cases before reaching for a formula.