Problem 69
Question
Find the \(11^{\text {th }}\) term of the arithmetic sequence \(\\{3 a-2 b, a+2 b,-a+6 b, \ldots\\}\)
Step-by-Step Solution
Verified Answer
The 11th term of the sequence is \(-17a + 38b\).
1Step 1: Identify the First Term
Examine the sequence \( \{3a - 2b, a + 2b, -a + 6b, \ldots \} \). The first term of this sequence is given by \( a_1 = 3a - 2b \).
2Step 2: Calculate the Common Difference
To find the common difference \(d\) of the arithmetic sequence, subtract the first term from the second term: \(d = (a + 2b) - (3a - 2b) = a + 2b - 3a + 2b = -2a + 4b \).
3Step 3: Use the Formula for the N-th Term
The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\). Here, we are looking for the \(11^{\text{th}}\) term: \(a_{11} = a_1 + 10d\).
4Step 4: Calculate the 11th Term
Substitute \(a_1 = 3a - 2b\) and \(d = -2a + 4b\) into the \(a_n\) formula: \(a_{11} = (3a - 2b) + 10(-2a + 4b)\).Simplifying, \(a_{11} = 3a - 2b - 20a + 40b\).Further simplification gives \(a_{11} = -17a + 38b\).
Key Concepts
Common DifferenceN-th Term FormulaSequence TermsSequence Calculation
Common Difference
An arithmetic sequence is defined by having a constant difference between any two consecutive terms. This difference is called the **common difference**.
To discover the common difference, subtract the first term from the second term in the sequence.
Let's consider the sequence: \(\{3a-2b, a+2b, -a+6b, \ldots\}\). You can find the common difference \(d\) by calculating \((a+2b) - (3a-2b)\). Simplifying this gives \(-2a + 4b\).
This result means that each term increases or decreases by \(-2a + 4b\), a consistent and predictable pattern.
To discover the common difference, subtract the first term from the second term in the sequence.
Let's consider the sequence: \(\{3a-2b, a+2b, -a+6b, \ldots\}\). You can find the common difference \(d\) by calculating \((a+2b) - (3a-2b)\). Simplifying this gives \(-2a + 4b\).
This result means that each term increases or decreases by \(-2a + 4b\), a consistent and predictable pattern.
N-th Term Formula
The **n-th term formula** of an arithmetic sequence is pivotal for finding any term. This formula is expressed as \(a_n = a_1 + (n-1)d\), where:
Knowing this formula is incredibly useful, as it provides a direct computation to understand sequences.
- \(a_1\) is the first term of the sequence,
- \(n\) is the term number,
- \(d\) is the common difference.
Knowing this formula is incredibly useful, as it provides a direct computation to understand sequences.
Sequence Terms
**Sequence terms** are the individual elements in a sequence of numbers. These terms follow a particular order and rule, which in arithmetic sequences, involves the common difference.
Consider the sequence: \(\{3a-2b, a+2b, -a+6b, \ldots\}\). Each term can be found using the sequence's rule.
In our example, the first term \(a_1\) is \(3a-2b\). Knowing the rule allows calculation of subsequent terms by consistently applying the common difference \(-2a + 4b\). This regular pattern helps predict terms effectively.
Consider the sequence: \(\{3a-2b, a+2b, -a+6b, \ldots\}\). Each term can be found using the sequence's rule.
In our example, the first term \(a_1\) is \(3a-2b\). Knowing the rule allows calculation of subsequent terms by consistently applying the common difference \(-2a + 4b\). This regular pattern helps predict terms effectively.
Sequence Calculation
**Sequence calculations** involve operations to find specific sequence terms. These calculations are based on arithmetic rules.
We can find the terms of an arithmetic sequence using the first term and the common difference.
For instance, to find the 11th term \(a_{11}\) of our sequence, we start with the first term \(3a-2b\) and use \(a_{11} = a_1 + 10d\).
Here, substitute \(a_1 = 3a-2b\) and \(d = -2a + 4b\) to find \(a_{11} = -17a + 38b\).
By applying these formulas, calculations are straightforward and accurate.
We can find the terms of an arithmetic sequence using the first term and the common difference.
For instance, to find the 11th term \(a_{11}\) of our sequence, we start with the first term \(3a-2b\) and use \(a_{11} = a_1 + 10d\).
Here, substitute \(a_1 = 3a-2b\) and \(d = -2a + 4b\) to find \(a_{11} = -17a + 38b\).
By applying these formulas, calculations are straightforward and accurate.
Other exercises in this chapter
Problem 68
Find the \(5^{\text {th }}\) term of the arithmetic sequence \(\\{9 b, 5 b, b, \ldots\\}\).
View solution Problem 68
What term in the sequence \(a_{n}=\frac{n^{2}+4 n+4}{2(n+2)}\) has the value 41 ? Verify the result.
View solution Problem 70
Calculate the first eight terms of the sequences \(a_{n}=\frac{(n+2) !}{(n-1) !}\) and \(b_{n}=n^{3}+3 n^{2}+2 n,\) and then make a conjecture about the relatio
View solution Problem 67
Consider the sequence defined by \(a_{n}=-6-8 n .\) Is \(a_{n}=-421\) a term in the sequence? Verify the result.
View solution