Arithmetic sequences

key notes:

Arithmetic Sequences

  1. Definition:
    • An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference (d).
  2. General Form: The n-th term of an arithmetic sequence can be expressed as:

Where:

  • an​ = n-th term
  • a1​ = first term
  • ddd = common difference
  • n = term number

3. Common Difference (d):

  • The common difference can be calculated as: d=an−an−1
  • This can also be found by subtracting any two consecutive terms in the sequence.

4. Examples:

  • For the sequence 2, 5, 8, 11, 14:
    • First term a1=2
    • Common difference d=3
    • 5th term a5=2+(5−1)×3=14

5. Finding Specific Terms:

  • To find a specific term in an arithmetic sequence, use the formula for an with the values of a1,d, and n.

Learn with an example

Type the missing number in this sequence:

115, 108, 101,______, 87

First, look for a pattern. Notice how each number is 7 less than the previous number:

115, 108, 101, __, 87

To make the pattern complete, the number 94 must go in the blank space.

Type the missing numbers in this sequence:

91, 85,______,______, 67, 61, 55

First, look for a pattern. Notice how each number is 6 less than the previous number:

91,    85,    __,    __,    67,    61,    55

To make the pattern complete, the numbers 79 and 73 must go in the blank spaces.

Type the missing number in this sequence:

64, 74, 84, 94,____, 114

First, look for a pattern. Notice how each number is 10 more than the previous number:

64, 74, 84, 94, __, 114

To make the pattern complete, the number 104 must go in the blank space.

Let’s practice: