Sequences and Series Help (page 2)
Introduction to Sequences and Arithmetic
A sequence is a series of terms in which each term in the series is generated using a rule. Each value in the sequence is a called a term. The rule of a sequence could be "each term is twice the previous term" or "each term is 4 more than the previous term."
Tip: The first term of a sequence is referred to as the first term (not the zeroth term).
An arithmetic sequence is a series of terms in which the difference between any two consecutive terms in the sequence is always the same. For example, 2, 6, 10, 14, 18, … is an arithmetic sequence. The difference between any two consecutive terms is always +4.
Each term in the following sequence is five less than the previous term. What is the next term in the sequence?
38, 33, 28, 23, …
To find the next term in the sequence, take the last term given and subtract 5, because the rule of the sequence is "each term in the sequence is five less than the previous term." You know that 23 – 5 = 18, so the next term in the sequence is 18.
When analyzing a sequence, try to find the mathematical operation that you can perform to get the next number in the sequence. Let's try an example. Look carefully at the following sequence:
2, 4, 8, 16, 32, …
Notice that each successive term is found by multiplying the prior term by 2: 2 × 2 = 4, 4 × 2 = 8, and so on. Because each term is multiplied by a constant number (2), there is a constant ratio between the terms. Sequences that have a constant ratio between terms are called geometric sequences.
A geometric sequence is a series of terms in which the ratio between any two consecutive terms in the sequence is always the same. For example, 1, 3, 9, 27, 81, … is a geometric sequence. The ratio between any two consecutive terms is always 1:3—each term is three times the previous term.
Each term in the following sequence is six times the previous term. What is the value of x?
2, 12, x, 432, …
To find the value of x, the third term in the sequence, multiply the second term in the sequence, 12, by 6, because every term is six times the previous term: (12)(6) = 72. The third term in the sequence is 72. You can check your answer by multiplying 72 by 6: (72)(6) = 432, the fourth term in the sequence.
Finidng a Specific Term in a Geometric Sequence
On an exam, you may be asked to determine a specific term in a sequence. Sometimes you will be asked for the 20th, 50th, or 100th term of a sequence. It would be unreasonable in many cases to evaluate that term, but you can represent that term with an expression.
Each term in the following sequence is four times the previous term. What is the hundredth term of the sequence?
3, 12, 48, 192, …
Write each term of the sequence as a product: 3 is equal to 40 × 3, 12 is equal to 41 × 3, 48 is equal to 42 × 3, and 192 is equal to 43 × 3. Each term in the sequence is equal to 4 raised to an exponent, multiplied by 3. For each term, the value of the exponent is one less than the position of the term in the sequence. The fourth term in the sequence, 192, is equal to 4 raised to one less than four (3), multiplied by 3. Therefore, the 100th term of the sequence is equal to 4 raised to one less than 100 (99), multiplied by 3. The hundredth term is equal to 499 × 3.
Let's say you are asked to find the 30th term of a geometric sequence like 2, 4, 8, 16, 32, …You could answer such a question by writing out 30 terms of a sequence, but this is an inefficient method. It takes too much time. Let's determine the formula:
First, let's evaluate the terms.
2, 4, 8, 16, 32, …
Term 1 = 2
Term 2 = 4, which is 2 × 2
Term 3 = 8, which is 2 × 2 × 2
Term 4 = 16, which is 2 × 2 × 2 × 2
You can also write out each term using exponents:
Term 1 = 2
Term 2 = 2 × 21
Term 3 = 2 × 22
Term 4 = 2 × 23
You can now write a formula:
Term n = 2 × 2n – 1
So, Term 30 = 2 × 230 – 1 = 2 × 229
General Formula for Geometric Sequences
The generic formula for a geometric sequence is Term n = a1 × rn – 1, where n is the term you are looking for, a1 is the first term in the series, and r is the ratio that the sequence increases by. In the preceding example, n = 30 (the 30th term), a1 = 2 (because 2 is the first term in the sequence), and r = 2 (because the sequence increases by a ratio of 2; each term is two times the previous term).
You can use the formula Term n = a1 × rn – 1 when determining a term in any geometric sequence.
Some sequences are defined by rules that are a combination of operations. The terms in these sequences do not differ by a constant value or ratio. For example, each number in a sequence could be generated by the rule "double the previous term and add one":
5, 11, 23, 47, 95, …
Each term in the following sequence is one less than four times the previous term. What is the next term in the sequence?
1, 3, 11, 43, …
Take the last given term in the sequence, 43, and apply the rule:
4(43) – 1 = 172 – 1 = 171
Find practice problems and solutions for these concepts at Sequences and Series Practice Problems.
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