Question
3. Consider the arithmetic sequence: \( -8 ;-2 ; 4 ; 10 ; \ldots \) 3.1 Write down the next term of the sequence 3.2 If the nth term of the sequence is 148 , determine the value of \( n \)
Ask by Chen Elliott. in South Africa
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**3.1. Next Term:** 16
**3.2. Value of \(n\):** 27
Solution
Let's analyze the given arithmetic sequence and solve the problems step by step.
**Given Sequence:** \(-8, -2, 4, 10, \ldots\)
### 3.1. Write Down the Next Term of the Sequence
In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term.
- **First Term (\(a_1\))**: \(-8\)
- **Common Difference (\(d\))**: \(-2 - (-8) = 6\)
To find the next term after \(10\):
\[
\text{Next Term} = 10 + d = 10 + 6 = 16
\]
**Answer:** The next term is **16**.
---
### 3.2. Determine the Value of \(n\) When the \(n^{th}\) Term is 148
The formula for the \(n^{th}\) term (\(a_n\)) of an arithmetic sequence is:
\[
a_n = a_1 + (n - 1) \cdot d
\]
Given:
- \(a_n = 148\)
- \(a_1 = -8\)
- \(d = 6\)
Plugging in the values:
\[
148 = -8 + (n - 1) \cdot 6
\]
\[
148 + 8 = 6(n - 1)
\]
\[
156 = 6(n - 1)
\]
\[
\frac{156}{6} = n - 1
\]
\[
26 = n - 1
\]
\[
n = 27
\]
**Answer:** The value of \(n\) is **27**.
Reviewed and approved by the UpStudy tutoring team
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Bonus Knowledge
The next term of the sequence can be found by observing that the pattern has a common difference. The difference between consecutive terms is \(6\) (from \(-8\) to \(-2\), \(-2\) to \(4\), and \(4\) to \(10\)). So, the next term after \(10\) is \(10 + 6 = 16\). To find \(n\) when the \(n\)th term is \(148\), we use the formula for the \(n\)th term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\), where \(a_1 = -8\) and \(d = 6\). Setting the equation \(148 = -8 + (n - 1) \cdot 6\) and solving for \(n\), we get \(n = 26\).