Respuesta :
Answer:
The formula is [tex]a_n = 3n-2[/tex]
The 14th term is 40, in other words, [tex]a_{14} = 40[/tex]
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Explanation:
The given sequence is 1, 4, 7, 10, ...
Subtract each adjacent term:
- 4-1 = 3
- 7-4 = 3
- 10-7 = 3
This shows that each term is increasing by 3. This is the common difference, so d = 3.
The first term is [tex]a_1 = 1[/tex]
The nth term of the arithmetic sequence is...
[tex]a_n = a_1 + d(n-1)\\\\a_n = 1 + 3(n-1)\\\\a_n = 1 + 3n-3\\\\a_n = 3n-2\\\\[/tex]
As a check, let's plug in say n = 3 to find that:
[tex]a_n = 3n-2\\\\a_3 = 3(3)-2\\\\a_3 = 9-2\\\\a_3 = 7\\\\[/tex]
The third term is 7, which matches with the sequence given to us. I'll let you confirm the other values.
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We'll repeat this idea but now for n = 14 to find the 14th term of the sequence.
[tex]a_n = 3n-2\\\\a_{14} = 3(14)-2\\\\a_{14} = 42-2\\\\a_{14} = 40\\\\[/tex]
The 14th term is 40.
You can make a table of values to help confirm the answer. In the first column you'll have the values of n (1,2,3,...) all the way up to 14. The second column will consist of the sequence 1, 4, 7, 10, ... each time going up by 3 until you reach the 14th term of 40.
