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Figurate numbers can be represented by an equally spaced arrangement of points around regular shapes such as squares and pentagons.
Example:
{49, 64, 81, 100, 121, ?, ...}
Answer: 144 (12x12).
Explanation: This sequence consists of squaring integers in order.
Example 2
{512, 343, 216, 125, ?, ...}
Answer: 43
64 = (4×4×4)
Explanation: Descending order of cubes.
1. Starting at the 0th triangular number, which is the fourth term?
Answer:6
Triangula number sequence: {0, 1, 3, 6, 10, 15 ...}
2. Starting at the 0th square number, which is the fifth term?
Answer:16
Squares number sequence: {0, 1, 4, 9, 16, 25...}
3. Starting at the first pentagonal number, which is the second term?
Answer: 5
Pentagonal number sequence: {1, 5, 12, 22, 35, 51, ...}
4. Starting at the first hexagonal number, which is the first term?
Answer: 1
Hexagonal number sequence: {1, 6, 15, 28, 45, 66, ...}
5. The formula for the nth hexagonal number is given by:
an = n(2n − 1)
Which is the fifth hexagonal number?
Answer: 45
Explanation: a5 = 5(10 − 1) = 45