Question

Which of the following explains the pattern below and makes the correct conjecture about the next two terms? -1, 0, 3, 8, 15,...

a) Adding consecutive prime numbers
b) Squaring consecutive odd numbers
c) Multiplying by consecutive even numbers
d) Adding consecutive square numbers

Answer 1

The pattern of the sequence -1, 0, 3, 8, 15,... is explained by (b) Squaring consecutive odd numbers.

Step-by-step explanation:

The sequence follows the pattern of squaring consecutive odd numbers. Starting with the first term
`\((-1)^2\)` equals 1, the second term
`\(0^2\)` equals 0, the third term is
`\(3^2\)` which equals 9, the fourth term is
`\(4^2\)` which equals 16, and so on. Each term in the sequence is obtained by squaring the consecutive odd numbers and subtracting 1. The general form of the pattern is
`\(n^2 - 1\)`, where n represents consecutive odd numbers.

For example:

- First term:
`\((-1)^2 - 1 = 1 - 1 = 0\)`

- Second term:
`\(0^2 - 1 = 0 - 1 = -1\)`

- Third term:
`\(3^2 - 1 = 9 - 1 = 8\)`

- Fourth term:
`\(4^2 - 1 = 16 - 1 = 15\)`

The next two terms in the sequence would be obtained by squaring the next consecutive odd numbers in the pattern:
`\(6^2 - 1 = 35\)` and
`\(7^2 - 1 = 48\)`. Therefore, the correct explanation for the pattern and the conjecture about the next two terms are based on squaring consecutive odd numbers.