Problem 1
Question
A geometric sequence is a sequence in which the ______ of successive terms is constant.
Step-by-Step Solution
Verified Answer
The ratio of successive terms is constant.
1Step 1: Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This means the ratio of the second term to the first term, the third term to the second term, and so on, remains constant.
2Step 2: Identify the Blank
The blank in the given statement is filled with the word that describes the mathematical operation that defines the relationship between terms in a geometric sequence. We have learned that this key operation is multiplication, specifically by the common ratio.
3Step 3: Complete the Sentence
The term we are looking for is 'ratio' because the defining feature of a geometric sequence is the constant ratio between successive terms. Hence, we fill the blank with the word 'ratio'.
Key Concepts
Common Ratio in Geometric SequencesUnderstanding Sequences and SeriesMathematical Operations in Geometric Sequences
Common Ratio in Geometric Sequences
In a geometric sequence, a central concept is the "common ratio." The common ratio is the fixed, non-zero number by which we multiply a term to get the subsequent term. This consistent factor determines how the sequence progresses. Thus, if you know the common ratio and the first term, you can calculate any term in the sequence.
For instance, in the sequence 2, 6, 18, 54, the common ratio is 3. We can find it by dividing the second term by the first term (\(6 \div 2 = 3\)) or the third term by the second term (\(18 \div 6 = 3\)). The entire sequence can be described using the initial term and this ratio. The formula for the \(n^{th}\) term in a geometric sequence can be expressed as:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where \(a_n\) is the \(n^{th}\) term, \(a_1\) is the first term, and \(r\) is the common ratio.
Rewrite the sequence using this formula to reinforce your understanding of how each term relates to the common ratio.
For instance, in the sequence 2, 6, 18, 54, the common ratio is 3. We can find it by dividing the second term by the first term (\(6 \div 2 = 3\)) or the third term by the second term (\(18 \div 6 = 3\)). The entire sequence can be described using the initial term and this ratio. The formula for the \(n^{th}\) term in a geometric sequence can be expressed as:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where \(a_n\) is the \(n^{th}\) term, \(a_1\) is the first term, and \(r\) is the common ratio.
Rewrite the sequence using this formula to reinforce your understanding of how each term relates to the common ratio.
Understanding Sequences and Series
Sequences are ordered lists of numbers following a specific pattern. In the case of a geometric sequence, this pattern involves multiplying by the common ratio. Identifying this pattern is crucial, as it allows us to predict the future terms of the sequence.
There are two types of sequences to be aware of:
\[ S_n = a_1 \frac{(1 - r^n)}{(1 - r)} \]
where \(S_n\) is the sum of the first \(n\) terms. This formula is valid when the common ratio \(r eq 1\). Understanding these formulas and their applications enhances comprehension of sequences and aids in solving problems effectively.
There are two types of sequences to be aware of:
- Finite sequences: These have a definite number of terms.
- Infinite sequences: These continue indefinitely.
\[ S_n = a_1 \frac{(1 - r^n)}{(1 - r)} \]
where \(S_n\) is the sum of the first \(n\) terms. This formula is valid when the common ratio \(r eq 1\). Understanding these formulas and their applications enhances comprehension of sequences and aids in solving problems effectively.
Mathematical Operations in Geometric Sequences
Mathematical operations are the core processes used to manipulate and understand sequences and series. In geometric sequences, multiplication is the key operation applied repetitively to derive successive terms using the common ratio. This multiplicative factor creates a systematic progression among the terms.
Sometimes, you might need to add or subtract terms when dealing with series. For example, finding the sum of several terms in a geometric series involves adding all these multiplied terms. You may also need to perform operations like division to verify the common ratio.
Here’s a breakdown of mathematical operations in geometric sequences:
Sometimes, you might need to add or subtract terms when dealing with series. For example, finding the sum of several terms in a geometric series involves adding all these multiplied terms. You may also need to perform operations like division to verify the common ratio.
Here’s a breakdown of mathematical operations in geometric sequences:
- Multiplication: Used to find the next term using the common ratio
- Division: Useful to calculate or verify the common ratio by dividing one term by its predecessor
- Addition: Utilized when computing the sum of a series
Other exercises in this chapter
Problem 1
An annuity is a sum of money that is paid in regular equal payments. The _____ of an annuity is the sum of all the individual payments together with all the int
View solution Problem 1
An algebraic expression of the form \(a+b,\) which consists of a sum of two terms, is called a _____.
View solution Problem 2
The \(n\) th partial sum of a sequence is the sum of the first __________ terms of the sequence. So for the sequence \(a_{n}=n^{2}\) the fourth partial sum is \
View solution Problem 2
The sequence given by \(a_{n}=a+(n-1) d\) is an arithmetic sequence in which \(a\) is the first term and \(d\) is the ____ . So for the arithmetic sequence \(a_
View solution