Evenly Spaced Sequences

To understand consecutive integers, first consider evenly spaced sequences. These are sequences of numbers whose values go up or down by the same amount (the increment) from one item in the sequence to the next. For instance, the sequence {4, 7, 10, 13, 16} is evenly spaced because each value increases by 3 over the previous value.

Consecutive multiples are special cases of evenly spaced sequences: all of the values in the sequence are multiples of the increment. For example, {12, 16, 20, 24} is a sequence of consecutive multiples because the values increase from one to the next by 4, and each element is a multiple of 4. Note that sequences of consecutive multiples must be composed of integers.

Consecutive integers are special cases of consecutive multiples: all of the values in the sequence increase by 1, and all integers are multiples of 1. For example, {12, 13, 14, 15, 16} is a sequence of consecutive integers because the values increase from one to the next by 1, and each element is an integer.

The relations among evenly spaced sequences, consecutive multiples, and consecutive integers are displayed in the following diagram.

• All sequences of consecutive integers are sequences of consecutive multiples.
• All sequences of consecutive multiples are evenly spaced sequences.
• All evenly spaced sequences are fully defined if the following three parameters are known:
  1. The smallest (first) or largest (last) number in the sequence
  2. The increment (always 1 for consecutive integers)
  3. The number of items in the sequence

Check Your Skills

1. Which of the following are evenly spaced sequences?

   1. 
   2. x, x − 4, x − 8, x − 12, x − 16
   3. 
   4. 5¹, 5², 5³, 5⁴, 5⁵
   5. y, 2y, 3y, 4y, 5y