Question
The twelfth term of an anithmetic sequence is 5 and the common difference of the sequence is 3 (a) Determine which term has a value of 47
Ask by Evans Chang. in South Africa
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The 26th term of the sequence is 47.
Solution
To determine which term of the arithmetic sequence has a value of **47**, let's follow these steps:
### Given:
- **Twelfth term (\(a_{12}\))**: 5
- **Common difference (\(d\))**: 3
### Formula for the \(n^{th}\) term of an arithmetic sequence:
\[
a_n = a_1 + (n - 1) \times d
\]
where:
- \(a_n\) is the \(n^{th}\) term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.
### Step 1: Find the first term (\(a_1\))
Given \(a_{12} = 5\) and \(d = 3\),
\[
a_{12} = a_1 + (12 - 1) \times 3
\]
\[
5 = a_1 + 33
\]
\[
a_1 = 5 - 33
\]
\[
a_1 = -28
\]
### Step 2: Find the term number (\(n\)) when \(a_n = 47\)
Using the formula:
\[
47 = -28 + (n - 1) \times 3
\]
\[
47 + 28 = (n - 1) \times 3
\]
\[
75 = 3(n - 1)
\]
\[
n - 1 = \frac{75}{3}
\]
\[
n - 1 = 25
\]
\[
n = 26
\]
### **Answer:**
The **26th term** of the arithmetic sequence is **47**.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find which term has a value of 47, we can use the formula for the nth term of an arithmetic sequence, given by \( a_n = a_1 + (n-1)d \). We know the twelfth term (\( a_{12} \)) is 5, which can be expressed as: \[ 5 = a_1 + (12-1) \cdot 3 \] This simplifies to: \[ 5 = a_1 + 33 \] \[ a_1 = 5 - 33 = -28 \] Now, to find which term equals 47, we set \( a_n = 47 \): \[ 47 = -28 + (n-1) \cdot 3 \] \[ 47 + 28 = (n-1) \cdot 3 \] \[ 75 = (n-1) \cdot 3 \] \[ n-1 = 25 \quad \Rightarrow \quad n = 26 \] Thus, the term that has a value of 47 is the 26th term. The answer is: The 26th term of the arithmetic sequence has a value of 47.
