Question

Write an explicit formula for the sequence 1,4,7,10... Then find a14.

Answer 1

Answer:

The formula is $a_n = 3n-2$

The 14th term is 40, in other words, $a_{14} = 40$

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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 $a_1 = 1$

The nth term of the arithmetic sequence is...

$a_n = a_1 + d(n-1)\\\\a_n = 1 + 3(n-1)\\\\a_n = 1 + 3n-3\\\\a_n = 3n-2$

As a check, let's plug in say n = 3 to find that:

$a_n = 3n-2\\\\a_3 = 3(3)-2\\\\a_3 = 9-2\\\\a_3 = 7$

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.

$a_n = 3n-2\\\\a_{14} = 3(14)-2\\\\a_{14} = 42-2\\\\a_{14} = 40$

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.