The nth even and odd number

The idea

The nth even number is 2n, and since each odd number is one less than the even number at the same position, the nth odd number is 2n – 1, where n is the position number.

The 100th odd number

What is the 100th even number? It is 2 × 100 = 200. What about the 100th odd number? Let us compare the two lists place by place.

The 100th odd number is…

• 199
• 201
• 100

Even and odd, place by place

Slide the position n. See the even number as n pairs. Build the rule for the nth even number, then for the nth odd number. A tick means the rule matches that column.

One less than the even number

At any position, the odd number is one less than the even number. The even number at position *n* is 2 × n.

To find an odd number at a given position: (a) find the even number there (2 times the position), (b) subtract 1.

The nth even number is 2n, and since each odd number is one less than the even number at the same position, the nth odd number is 2n − 1, where *n* is the position number.

• Position | Even: 2n | Odd: 2n − 1
• 1 | 2 | 1
• 2 | 4 | 3
• 3 | 6 | 5
• 100 | 200 | 199

Notes

The nth even number is 2n, and since each odd number is one less than the even number at the same position, the nth odd number is 2n − 1, where *n* is the position number.

Check yourself

What is the 50th odd number?

Answer: 99

2 × 50 − 1 = 100 − 1 = 99.

Which formula gives the nth odd number?

In the list of odd numbers 1, 3, 5, 7, … at which position is the number 41?

Answer: 21

41 = 2n − 1 means 2n = 42, so n = 21.

Which statement is true at every position *n*?

• 2n + 1. For n = 1 it gives 3, but the first odd number is 1. It gives the odd number one place further on.
• 2n. That is the nth even number.
• n − 1. For n = 5 this gives 4, which is not odd.
• 2n − 1 — correct. Yes! For n = 1, 2, 3 it gives 1, 3, 5.
• The odd number is 1 less than the even number — correct. Yes! 2n − 1 is one less than 2n.
• The odd number is 1 more than the even number. At position 3 the even number is 6 and the odd number is 5: one less, not more.
• The odd number is half the even number. At position 3 the even number is 6 but the odd number is 5, not 3.
• The odd number is the same as the position n. At position 3 the odd number is 5, not 3.