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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: 4^{3}

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 n^{th} hexagonal number is given by:

a_{n} = n(2n − 1)

Which is the fifth hexagonal number?

Answer: 45

Explanation: a_{5} = 5(10 − 1) = 45