KA-0107Growth patterns and figurate numbers
4 points, difficulty 2 of 3Level 5–6about 90sDots are arranged in triangles: 1 dot, then 3, then 6, then 10, and so on. How many dots does the tenth triangle have?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
What is added from one triangle to the next?
2The strategy
The jumps are not equal: each new triangle adds one more row than the last one added. Write the differences down and you will see the pattern in them.
3The full solution
The differences are 2, 3, 4, 5 and so on, so the tenth triangle is 1 + 2 + 3 + ... + 10 = 55.
Solution
The reliable way
Look at what is added: 1 to 3 adds 2, 3 to 6 adds 3, 6 to 10 adds 4. Each step adds one more than the last, because each new triangle gains a longer bottom row. So the tenth triangle is 1 + 2 + 3 + ... + 10, which pairs up as five lots of 11 = 55. The transferable idea: when the differences themselves form a pattern, add the differences rather than continuing the drawing.
The elegant way
The nth triangular number is n times n plus one, halved: 10 x 11 / 2 = 55.
Why this is on the test: Growth patterns are the first place a constant difference fails, and noticing that the differences themselves grow is the whole move.