Answer :
Final answer:
The number 163 is a term in the arithmetic sequence 10, 13, 16, 19,... because it is the 52nd term in the sequence. This sequence has a common difference of 3. The formula for the n-th term confirms this result.
Explanation:
Determining if 163 is a Term in the Sequence
The sequence given is: 10, 13, 16, 19.... This is an arithmetic sequence where each term increases by a constant difference. To find the common difference, we can subtract the first term from the second:
- 13 - 10 = 3
- 16 - 13 = 3
- 19 - 16 = 3
The common difference in this sequence is 3.
We can represent the n-th term of an arithmetic sequence using the formula:
a_n = a_1 + (n - 1) d
where:
- a_n = n-th term
- a_1 = first term (10 in this case)
- d = common difference (3)
- n = term number
Substituting the known values into the formula gives:
a_n = 10 + (n - 1) 3
Now, to determine if 163 is a term in the sequence, we need to set a_n = 163 and solve for n:
163 = 10 + (n - 1) 3
Subtracting 10 from both sides:
153 = (n - 1) 3
Now divide both sides by 3:
51 = n - 1
Add 1 to both sides:
n = 52
This indicates that 163 is indeed the 52nd term in the sequence. Therefore, the answer is:
Yes, 163 is a term in the sequence.
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