Question

Multiply each odd number by 2 and add 1. If these numbers are written in order, does it form and arithmetic sequence? Write the algebraic form of this sequence. Now consider the odd numbers not in this sequence. Do they form and arithmetic sequence? What is its algebraic form?

Answer 1

The odd numbers are 1, 3, 5, …

When we multiply each odd number by 2 and add 1, then the sequence becomes:

(1 × 2) + 1, (3 × 2) + 1, (5 × 2) + 1, …

Or 3, 7, 11, …

Difference between second and first term = 7 − 3 = 4

Difference between third and second term = 11 − 7 = 4

Since the difference between the consecutive terms is same, the sequence is an arithmetic sequence with common difference = a = 4.

First term = a + b = 3

⇒ 4 + b = 3

⇒ b = 3 − 4 = −1

∴

_{Thus, the algebraic form of the given sequence is 4n − 1.}

The odd numbers not in the above sequence are 1, 5, 9, 13, …

Difference between second and first term = 5 − 1 = 4

Difference between third and second term = 9 − 5 = 4

Since the difference between the consecutive terms is same, the sequence is an arithmetic sequence with common difference = a = 4.

First term = a + b = 1

⇒ 4 + b = 1

⇒ b = 1 − 4 = −3

∴

_{Thus, the algebraic form of the given sequence is 4n − 3.}