Problem:
Find the 5th term of the arithmetic sequence with a_8 = 59 and a_16 = 107.
Given:
The nth term of an arithmetic sequence is given by: a_n = a_1 + (n - 1)d
Where: a_1 = first term
- 5 = number of the terms
- d = common difference
- a_5 = last term or nth term
Formula:
To find the 5th term we need to use the arithmetic sequence formula:
Solution:
First, find the common difference.
- a_n = a_1 + (n - 1)d
- a_9 = a_1 + (n - 1)d
- 107 = 59 + (9 - 1)d
- 107 = 59 + (8)d
- 107 + (-59) = 59 + (-59) + 8d
- 48 = 8d
- 8d/8 = 48/8
- d = 6
Next, find the a_1.
- a_n = a_1 + (n - 1)d
- a_8 = a_1 + (n - 1)d
- 59 = a_1 + (8 - 1)6
- 59 = a_1 + (7)6
- 59 = a_1 + 42
- 59 + (-42) = a_1 + 42 + (-42)
- a_1 = 17
Lastly, find the a_5.
- a_n = a_1 + (n - 1)d
- a_5 = a_1 + (n - 1)d
- a_5 = 17 + (5 - 1)6
- a_5 = 17 + (4)6
- a_5 = 17 + 24
- a_5 = 41
Answer:
Therefore, the 5th term of the arithmetic sequence is 41.
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