Problem 29
Question
In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=-20, d=-4$$
Step-by-Step Solution
Verified Answer
The general term of the sequence is given by the formula \(a_n = -4n -16\). The 20th term of the sequence is -96.
1Step 1: Write down the formula for an arithmetic sequence
We start by using the standard general term formula of an arithmetic sequence, which is \(a_n = a_1 + (n-1) * d\). Here, \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
2Step 2: Substitute the values into the general term formula
Now we substitute the given values into our formula. \(a_1 = -20, d = -4\), so the formula becomes \(a_n = -20 + (n-1)* -4\).
3Step 3: Simplify the formula
Simplify the equation to obtain the formula for any term n by distributing the -4 into the parentheses to yield \(a_n = -20 -4n + 4\). Combine like terms to get \(a_n = -4n -16\).
4Step 4: Find the 20th term using the formula
We use the derived formula \(a_n = -4n -16\) to calculate the 20th term. Substituting n = 20 into the formula gives \(a_{20} = -4*20 -16 = -80 -16 = -96\).
Key Concepts
Arithmetic SequenceGeneral TermCommon DifferenceSequence Simplification
Arithmetic Sequence
An arithmetic sequence is an ordered list of numbers where the difference between consecutive terms is constant. This constant difference is what defines the sequence and is known as the common difference. For example, if we begin with the number 2 and keep adding 3, we'll get an arithmetic sequence: 2, 5, 8, 11, and so on.
An important property of arithmetic sequences is their predictability, which allows us to calculate any term within the sequence without listing out all the preceding terms. This feature becomes particularly useful when dealing with large sequences or trying to find specific terms far down the line, such as the 20th or 100th term.
An important property of arithmetic sequences is their predictability, which allows us to calculate any term within the sequence without listing out all the preceding terms. This feature becomes particularly useful when dealing with large sequences or trying to find specific terms far down the line, such as the 20th or 100th term.
General Term
The general term of an arithmetic sequence represents the nth term and is denoted by the symbol \( a_n \). It's a formula that expresses any term in the sequence in terms of the initial term, the common difference, and the term's position within the sequence. The standard formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( d \) is the common difference between the terms, and \( n \) is the term number.
To find the 20th term of a sequence, as in our exercise, we would use this formula with \( n = 20 \). Let's say the first term of our sequence (\( a_1 \)) is 3 and the common difference (\( d \)) is 5; the 20th term would be calculated by plugging these values into our formula: \( a_{20} = 3 + (20-1) \times 5 \).
To find the 20th term of a sequence, as in our exercise, we would use this formula with \( n = 20 \). Let's say the first term of our sequence (\( a_1 \)) is 3 and the common difference (\( d \)) is 5; the 20th term would be calculated by plugging these values into our formula: \( a_{20} = 3 + (20-1) \times 5 \).
Common Difference
In the realm of arithmetic sequences, the term 'common difference' refers to the consistent interval or difference between consecutive terms and is represented by the symbol \( d \). This difference can be positive, negative, or even zero, leading to an increasing, decreasing, or constant sequence, respectively.
Knowing the common difference is crucial as it essentially sets the pattern of the entire sequence. For instance, if our sequence starts with 6 and the common difference is -2, the next term would be 4, followed by 2, 0, and so on, consistently decreasing by 2.
Knowing the common difference is crucial as it essentially sets the pattern of the entire sequence. For instance, if our sequence starts with 6 and the common difference is -2, the next term would be 4, followed by 2, 0, and so on, consistently decreasing by 2.
Sequence Simplification
Sequence simplification is a process that involves algebraically manipulating the general term formula to make it more straightforward to work with, especially for calculating specific terms. This often involves distributing the common difference, merging like terms, and rewriting the formula in its simplest form.
For example, given the general formula \( a_n = a_1 + (n-1)d \) and the values \( a_1 = -20 \) and \( d = -4 \), we simplify this as explained in our exercise solution. Distributing the common difference across the \( n-1 \) term and then combining like terms results in the simplified formula \( a_n = -4n - 16 \), which is much easier to use for calculating specific terms within the sequence, such as the 20th term.
For example, given the general formula \( a_n = a_1 + (n-1)d \) and the values \( a_1 = -20 \) and \( d = -4 \), we simplify this as explained in our exercise solution. Distributing the common difference across the \( n-1 \) term and then combining like terms results in the simplified formula \( a_n = -4n - 16 \), which is much easier to use for calculating specific terms within the sequence, such as the 20th term.
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