What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, ... ? Which sequence do you get? Why? Can you explain it with a picture?
When you add pairs of consecutive triangular numbers, you get the Square numbers sequence:
1 + 3 = 4 3 + 6 = 9 6 + 10 = 16 10 + 15 = 25
This gives: 4, 9, 16, 25, ... which are 2², 3², 4², 5², ...
Pictorial Explanation: Two consecutive triangular numbers can be arranged together to form a square.
The nth triangular number plus the (n+1)th triangular number equals (n+1)².
Example: The triangular numbers 3 (2nd triangular) and 6 (3rd triangular) can be arranged as a square: One triangle of 3 dots plus another triangle of 6 dots combine to form 9 dots in a square.
Explanation
The triangular numbers are 1, 3, 6, 10, 15, ... Adding consecutive pairs: 1+3=4=2², 3+6=9=3², 6+10=16=4², 10+15=25=5². This gives square numbers starting from 2². The pictorial explanation involves combining two triangular arrangements of dots to form a square arrangement. The nth triangular number T(n) = n(n+1)/2, and T(n) + T(n+1) = n(n+1)/2 + (n+1)(n+2)/2 = (n+1)².
Solution Steps
Identify the triangular numbers sequence: 1, 3, 6, 10, 15, ...
Add consecutive pairs: 1+3=4, 3+6=9, 6+10=16, 10+15=25
Recognize the result as square numbers: 4, 9, 16, 25, ... = 2², 3², 4², 5², ...
Understand pictorially: Two triangular arrangements combine to form a square