Chapter12: Sequences, Series And Binomial Theorem
Section: Chapter Questions
Problem 327PT: Find the twenty-third term of an arithmetic sequence whose seventh term is 11 and common difference...
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Question
![### Arithmetic Sequence Partial Sum Calculation
**Question:**
What is the partial sum of terms 12 to 15 of the arithmetic sequence 88, 75, 62, 49, ...?
**Answer Choices:**
- O 198
- O -45
- O -94
- O -298
To solve this problem, follow these steps:
1. **Identify the first term (a) and the common difference (d) of the sequence:**
- The first term (a) is 88.
- The common difference (d) is calculated as:
\[
d = 75 - 88 = -13
\]
2. **Determine the formula for the nth term of the arithmetic sequence:**
\[
a_n = a + (n-1) \cdot d
\]
3. **Calculate the terms from 12 to 15:**
- For \(a_{12}\):
\[
a_{12} = 88 + (12 - 1) \cdot (-13) = 88 + 11 \cdot (-13) = 88 - 143 = -55
\]
- For \(a_{13}\):
\[
a_{13} = 88 + (13 - 1) \cdot (-13) = 88 + 12 \cdot (-13) = 88 - 156 = -68
\]
- For \(a_{14}\):
\[
a_{14} = 88 + (14 - 1) \cdot (-13) = 88 + 13 \cdot (-13) = 88 - 169 = -81
\]
- For \(a_{15}\):
\[
a_{15} = 88 + (15 - 1) \cdot (-13) = 88 + 14 \cdot (-13) = 88 - 182 = -94
\]
4. **Sum the terms from 12 to 15:**
\[
S = a_{12} + a_{13} + a_{14} + a_{15} = -55 + (-68) + (-81) + (-94) = -298
\]
The correct answer to the question is:
- O -298](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3047b756-b2f4-4f66-ba10-7c0773ac07fd%2Fa9bbd385-bbc7-4f74-8938-fb4da03e7eca%2Fet7qi5_processed.png&w=3840&q=75)
Transcribed Image Text:### Arithmetic Sequence Partial Sum Calculation
**Question:**
What is the partial sum of terms 12 to 15 of the arithmetic sequence 88, 75, 62, 49, ...?
**Answer Choices:**
- O 198
- O -45
- O -94
- O -298
To solve this problem, follow these steps:
1. **Identify the first term (a) and the common difference (d) of the sequence:**
- The first term (a) is 88.
- The common difference (d) is calculated as:
\[
d = 75 - 88 = -13
\]
2. **Determine the formula for the nth term of the arithmetic sequence:**
\[
a_n = a + (n-1) \cdot d
\]
3. **Calculate the terms from 12 to 15:**
- For \(a_{12}\):
\[
a_{12} = 88 + (12 - 1) \cdot (-13) = 88 + 11 \cdot (-13) = 88 - 143 = -55
\]
- For \(a_{13}\):
\[
a_{13} = 88 + (13 - 1) \cdot (-13) = 88 + 12 \cdot (-13) = 88 - 156 = -68
\]
- For \(a_{14}\):
\[
a_{14} = 88 + (14 - 1) \cdot (-13) = 88 + 13 \cdot (-13) = 88 - 169 = -81
\]
- For \(a_{15}\):
\[
a_{15} = 88 + (15 - 1) \cdot (-13) = 88 + 14 \cdot (-13) = 88 - 182 = -94
\]
4. **Sum the terms from 12 to 15:**
\[
S = a_{12} + a_{13} + a_{14} + a_{15} = -55 + (-68) + (-81) + (-94) = -298
\]
The correct answer to the question is:
- O -298
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