Munoz Simpson
06/16/2024 · High School
7.20 (ii) \( -4,-7,-10, \cdots, a_{19} \) (iv) \( 9,14,19, \cdots, a_{14} \)
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Tutor-Verified Answer
Step-by-step Solution
To find the terms \( a_{19} \) and \( a_{14} \) for the given sequences, we first need to identify the patterns in each sequence.
### Sequence (ii): \( -4, -7, -10, \cdots, a_{19} \)
This sequence is an arithmetic sequence where the first term \( a_1 = -4 \) and the common difference \( d \) can be calculated as follows:
\[
d = -7 - (-4) = -3
\]
The general formula for the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]
Substituting the values we have:
\[
a_n = -4 + (n-1)(-3)
\]
To find \( a_{19} \):
\[
a_{19} = -4 + (19-1)(-3)
\]
\[
= -4 + 18 \cdot (-3)
\]
\[
= -4 - 54
\]
\[
= -58
\]
Thus, \( a_{19} = -58 \).
### Sequence (iv): \( 9, 14, 19, \cdots, a_{14} \)
This sequence is also an arithmetic sequence where the first term \( a_1 = 9 \) and the common difference \( d \) can be calculated as follows:
\[
d = 14 - 9 = 5
\]
Using the same formula for the \( n \)-th term:
\[
a_n = a_1 + (n-1) \cdot d
\]
Substituting the values we have:
\[
a_n = 9 + (n-1)(5)
\]
To find \( a_{14} \):
\[
a_{14} = 9 + (14-1)(5)
\]
\[
= 9 + 13 \cdot 5
\]
\[
= 9 + 65
\]
\[
= 74
\]
Thus, \( a_{14} = 74 \).
### Summary of Results
- \( a_{19} = -58 \)
- \( a_{14} = 74 \)
Quick Answer
\( a_{19} = -58 \), \( a_{14} = 74 \)
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