Sequences

General Information

You will occasionally be given problems that ask you to identify terms in a particular sequence. You will need to understand in particular how Arithmetic Sequences and Geometric Sequences. (There are of course other sequences that you will encounter, but these will have rules that are given to you at the time the sequences are introduced.)

Arithmetic Sequences

Arithmetic Sequences are sequences in which you add the same value (called the common difference) to get from one term to the next. As a result, between any two consecutive terms in an arithmetic sequence, the difference/gap will be the same. Below are two different equations for finding any term in an arithmetic sequence, where n is the sequence term number, a_n is the n^th term in the sequence, and d is the common difference.

• a_n = a₁ + d(n - 1)
• a_n = a₀ + dn

Though you will often see the first of these equations used to define an arithmetic sequence, both of them technically work. The zeroth term does not actually exist, but if you go backwards to what would have been the zeroth term, the second equation is much simpler and resembles the slope-intercept form of a line. (The common difference is the slope, and the zeroth term is the y-intercept.) Though arithmetic sequences are simple sequence types, they are sometimes easy to miss. Below are a few examples of arithmetic sequences, with the common difference in parentheses.

• 1, 2, 3, 4, 5, 6, 7, ... (1)
• 10, 9, 8, 7, 6, 5, ... (-1)
• ¹⁄₂, 1, ³⁄₂, 2, ⁵⁄₂, 3, ⁷⁄₂, 4, ... (¹⁄₂)
• 29, 29, 29, 29, 29, 29, 29, ... (0)

There are two short-cut features that you will want to remember about arithmetic sequences. The first is that the mean and median of an arithmetic sequence are always identical. This feature of the arithmetic sequence is due to the symmetry of sequence caused by the common difference. This first short-cut then leads to the second short-cut: if you can identify the mean/median of an arithmetic sequence, you can find the sum of those terms by multiplying that mean/median by the number of terms in the sequence. Let's look at the sequences listed above.

• 1, 2, 3, 4, 5, 6, 7 - mean/median = 4 - sum = 7*4 = 28
• 10, 9, 8, 7, 6, 5 - mean/median = 7.5 - sum = 6*7.5 = 45
• ¹⁄₂, 1, ³⁄₂, 2, ⁵⁄₂, 3, ⁷⁄₂, 4 - mean/median = ⁹⁄₄ = 2.25 - sum = 8*2.25 = 18
• 29, 29, 29, 29, 29, 29, 29 - mean/median = 29 - 7*29 = 203

Geometric Sequences

Geometric Sequences are sequences in which you multiply the same value (called the common ratio) to get from one term to the next. As a result, if you divide any term in the sequence by the previous term in the sequence, the value will always be the same. Below are two different equations for finding any term in a geometric sequence, where n is the sequence term number, a_n is the n^th term in the sequence, and r is the common ratio. Of note concerning geometric sequences is that they can alternate positive to negative and negative to positive. Arithmetic sequences can only go in one direction.

• a_n = a₁r^{n - 1}
• a_n = a₀rⁿ

As with the arithmetic sequence, the geometric sequence formula that will generally be given to you is the first of the two listed. However, even though the zeroth term does not technically exist, a phantom zeroth term can be calculated and used to make this formula simpler. Below are a few examples of geometric sequences, with common ratios in parentheses.

• 1, 2, 4, 8, 16, 32, ... (2)
• 32, 16, 8, 4, 2, 1, ... (¹⁄₂)
• 1, -2, 4, -8, 16, -32, ... (-2)
• 32, -16, 8, -4, 2, -1, ... (-¹⁄₂)
• 29, 29, 29, 29, 29, 29, 29, ... (1)

The main feature about geometric sequences to remember is that the sum of an infinite geometric sequence can be calculated if |r| < 1. This sum is calculate by using the equation ^a₁⁄_{1 - r}. An infinite geometric sequence will not have a finite sum if |r| is not less than 1.