Sequences
12.1 Logic
 
12.2 Sequences
 
12.3 Limits
 
12.4 Imaginary and Complex Numbers
 
 
12.5 Key Formulas
 
12.6 Review Questions
 
12.7 Explanations
 
Sequences
You might see one or two sequence questions on the Math IC. The two types of sequences tested are arithmetic and geometric sequences.
Arithmetic Sequences
An arithmetic sequence is an ordered list of terms in which the difference between consecutive terms is constant. In other words, the same value or variable is added to each term in order to create the next term: if you subtract any two consecutive terms of the sequence, you will get the same difference. An example is {an} = 1, 4, 7, 10, 13, ..., where 3 is the constant increment between values.
The notation of an arithmetic sequence is
where an is the nth term of the sequence and d is the difference between consecutive terms. For the Math IC, you must first be able to determine that a given sequence is an arithmetic sequence. To figure this out, take two sets of consecutive terms and subtract the smaller term from the larger. If the difference between the terms in the two sets is equal, you’ve got an arithmetic sequence. To determine if the sequence {an} = 1, 4, 7, 10, 13, … is arithmetic, take two sets of consecutive terms {1, 4} and {10, 13}, and subtract the first from the second:
Since the difference is equal, you know this sequence is arithmetic. You should be able to do three things with an arithmetic sequence:
  1. Find d
  2. Find the nth term
  3. Calculate the sum of the first n terms
Finding d
To find the difference, d, between the terms of an arithmetic sequence, just subtract one term from the next term. For the arithmetic sequence an = 1, 4, 7, 10, 13, … , . Here’s a slightly more complicated form of this question:
If a4 = 4 and a7 = 10, find d.
This question gives you the fourth and seventh terms of a sequence:
Since in arithmetic sequences d is constant between every term, you know that d + 4 = a5, a5 + d = a6, and a6 + d = 10. In other words, the difference between the seventh term, 10, and the fourth term, 4, is 3d. Stated as an equation:
Solving this equation is a process of simple algebra.
Finding the nth Term
To find the nth term in an arithmetic sequence, use the following formula:
In the example above, to find the 55th term we would have to find the value of a1 first. Plug the values of a4 = 4, n = 4 and d = 2 into the formula an = a1 + (n – 1)d to find that a1 equals –2. Now find the 55th term, a55 = –2 + (55 – 1)2 = –2 + (54)2 = –2 + 108 = 106.
Calculating the Sum of the First n Terms
In order to find the sum of the first n terms, simply find the value of the average term and then multiply that average by the number of terms you are summing.
As you can see, this is simply n times the average of the first n terms. Using the same example, the sum of the first 55 terms would be:
Geometric Sequences
A geometric sequence is a sequence in which the ratio of any term and the next term is constant. Whereas in an arithmetic sequence the difference between consecutive terms is always constant, in a geometric sequence the quotient of consecutive terms is always constant. The constant factor by which the terms of a geometric function differ is called the common ratio of the geometric sequence. The common ratio is usually represented by the variable r. Here is an example of a geometric sequence in which r = 3.
The general form of a geometric sequence is:
You should be able to identify a geometric sequence from its terms, and you should be able to perform three tasks on geometric sequences:
  1. Find d
  2. Find the nth term
  3. Calculate the sum of the first n terms
Finding r
To find the common ratio of a geometric sequence, all you have to do is divide one term by the preceding term.
For example, the value of r for the sequence 3, 6, 12, 24, ... is 63 = 2.
Finding the nth Term
To find the nth term of a geometric sequence, use the following formula:
For example, the 11th term of the sequence above is:
Calculating the Sum of the First n Terms
To find the sum of the first n terms of a geometric sequence, use the following formula:
So the sum of the first 10 terms of the same sequence is:
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