Problem 32
Question
For the following exercises, write the first five terms of the geometric sequence. $$ a_{n}=-4 \cdot 5^{n-1} $$
Step-by-Step Solution
Verified Answer
The first five terms are -4, -20, -100, -500, -2500.
1Step 1: Understand the Given Formula
The given formula for the geometric sequence is \( a_{n} = -4 \cdot 5^{n-1} \). This formula helps us find the terms in the sequence by substituting the values of \( n \). Here, \( a_1 \) is the first term, \( a_2 \) is the second term, and so on.
2Step 2: Calculate the First Term
To find the first term \( a_1 \), substitute \( n = 1 \) into the formula: \( a_1 = -4 \cdot 5^{1-1} = -4 \cdot 5^0 = -4 \cdot 1 = -4 \).
3Step 3: Calculate the Second Term
For the second term \( a_2 \), substitute \( n = 2 \) into the formula: \( a_2 = -4 \cdot 5^{2-1} = -4 \cdot 5^1 = -4 \cdot 5 = -20 \).
4Step 4: Calculate the Third Term
For the third term \( a_3 \), substitute \( n = 3 \) into the formula: \( a_3 = -4 \cdot 5^{3-1} = -4 \cdot 5^2 = -4 \cdot 25 = -100 \).
5Step 5: Calculate the Fourth Term
For the fourth term \( a_4 \), substitute \( n = 4 \) into the formula: \( a_4 = -4 \cdot 5^{4-1} = -4 \cdot 5^3 = -4 \cdot 125 = -500 \).
6Step 6: Calculate the Fifth Term
For the fifth term \( a_5 \), substitute \( n = 5 \) into the formula: \( a_5 = -4 \cdot 5^{5-1} = -4 \cdot 5^4 = -4 \cdot 625 = -2500 \).
Key Concepts
Sequence TermsGeometric ProgressionMathematical Formulas
Sequence Terms
A sequence is a list of numbers that are arranged in a specific order. In the context of a geometric sequence, each number in this list is known as a term. Each term is defined by its position in the sequence, such as the first term, second term, etc. Understanding the notion of sequence terms is crucial because it allows us to identify and analyze patterns within geometric progressions.
For example, when we refer to the first term, we use the notation \(a_1\). Likewise, for the second term, it’s \(a_2\), and so forth. These terms are connected through a specific formula that dictates how each term is calculated, given its position in the sequence.
In a geometric sequence, each term is generated using a fixed number, often called the common ratio, applied consistently to each prior term. This way, anyone can find the terms by following the pattern laid out by the initial term and the common ratio.
For example, when we refer to the first term, we use the notation \(a_1\). Likewise, for the second term, it’s \(a_2\), and so forth. These terms are connected through a specific formula that dictates how each term is calculated, given its position in the sequence.
In a geometric sequence, each term is generated using a fixed number, often called the common ratio, applied consistently to each prior term. This way, anyone can find the terms by following the pattern laid out by the initial term and the common ratio.
Geometric Progression
A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by a constant non-zero number known as the common ratio. This type of progression is distinct from arithmetic progression, where the difference between terms is constant.
In the exercise we analyzed, the geometric sequence follows the formula \(a_n = -4 \cdot 5^{n-1}\). In this formula:
Understanding geometric progressions is important as they are widely used in various real-world scenarios, such as calculating interest, population growth, and other exponential growth situations.
In the exercise we analyzed, the geometric sequence follows the formula \(a_n = -4 \cdot 5^{n-1}\). In this formula:
- \(-4\) is the first term of the sequence, or \(a_1\).
- \(5\) is the common ratio. It is the number each term is multiplied by to get the next term.
- \(n-1\) is the exponent applied to the common ratio to sequentially calculate each term.
Understanding geometric progressions is important as they are widely used in various real-world scenarios, such as calculating interest, population growth, and other exponential growth situations.
Mathematical Formulas
Mathematical formulas are expressions that help to succinctly represent relationships between different mathematical quantities. In the study of sequences, particularly geometric sequences, formulas are essential for accurately determining the progression of terms.
The formula \(a_n = -4 \cdot 5^{n-1}\) defines a geometric sequence where:
Using mathematical formulas not only streamlines computations but also provides a clear, logical path to follow when determining unknown values within a sequence. Mastery of these formulas enhances problem-solving skills in both academic and applied mathematics.
The formula \(a_n = -4 \cdot 5^{n-1}\) defines a geometric sequence where:
- \(a_n\) represents the \(n\)-th term in the sequence.
- \(-4\) is the starting value or first term when \(n = 1\).
- \(5\) is the common ratio.
Using mathematical formulas not only streamlines computations but also provides a clear, logical path to follow when determining unknown values within a sequence. Mastery of these formulas enhances problem-solving skills in both academic and applied mathematics.
Other exercises in this chapter
Problem 32
For the following exercises, find the distinct number of arrangements. The letters in the word "academia" that begin and end in "a"
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For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The third term of \((6 x-3 y)^{7}\)
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For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\\{-15,-7,1, \ldots\\} $$
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For the following exercises, write the first eight terms of the sequence. $$ a_{1}=-1, a_{2}=5, a_{n}=a_{n-2}\left(3-a_{n-1}\right) $$
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