Question
Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?
✓
Answer
Given table is as
1, 1, 1, 1, 1, 1, ... (All 1's)
1, 2, 3, 4, 5, 6, 7, ... (Counting numbers)
1, 3, 5, 7, 9, 11, 13, ... (Odd numbers)
2, 4, 6, 8, 10, 12, 14, ... (Even numbers)
1, 3, 6, 10, 15, 21, 28, ... (Triangular numbers)
1, 4, 9, 16, 25, 36, 49, ... (Squares)
1, 8, 27, 64, 125, 216, ... (Cubes)
1, 2, 3, 5, 8, 13, 21, ... (Virahanka numbers)
1, 2, 4, 8, 16, 32, 64, ... (Powers of 2)
1, 3, 9, 27, 81, 243, 729, ... (Powers of 3)
From the above table, we see that
On adding the counting numbers up and turn down we get the square numbers.
and so on.
Thus, counting number is relate with square numbers.
Also, we see triangular numbers added together for square numbers.
Also, we can look following patterns,
Here, we get a sequence of square of triangular numbers.
Thus, we can say that sequence relate to each other.
1, 1, 1, 1, 1, 1, ... (All 1's)
1, 2, 3, 4, 5, 6, 7, ... (Counting numbers)
1, 3, 5, 7, 9, 11, 13, ... (Odd numbers)
2, 4, 6, 8, 10, 12, 14, ... (Even numbers)
1, 3, 6, 10, 15, 21, 28, ... (Triangular numbers)
1, 4, 9, 16, 25, 36, 49, ... (Squares)
1, 8, 27, 64, 125, 216, ... (Cubes)
1, 2, 3, 5, 8, 13, 21, ... (Virahanka numbers)
1, 2, 4, 8, 16, 32, 64, ... (Powers of 2)
1, 3, 9, 27, 81, 243, 729, ... (Powers of 3)
From the above table, we see that
On adding the counting numbers up and turn down we get the square numbers.
and so on.
Thus, counting number is relate with square numbers.
Also, we see triangular numbers added together for square numbers.
Also, we can look following patterns,
Here, we get a sequence of square of triangular numbers.
Thus, we can say that sequence relate to each other.
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