What is the eighth term of an arithmetic sequence with a first term of 10 and a third term of 22?

In an arithmetic sequence, each term can be expressed in terms of the first term and the common difference. The formula for the nth term of an arithmetic sequence is given by:

[ a_n = a_1 + (n - 1) d ]

where ( a_n ) is the nth term, ( a_1 ) is the first term, ( d ) is the common difference, and ( n ) is the term number.

In this sequence, we know the first term ( a_1 ) is 10. The third term ( a_3 ) is given as 22. Using the formula for the third term:

[ a_3 = a_1 + (3 - 1) d ]

Substituting the known values:

[ 22 = 10 + 2d ]

To find the common difference ( d ), we can rearrange this equation:

[ 22 - 10 = 2d ]

[ 12 = 2d ]

[ d = 6 ]

Now that we have the common difference, we can find the eighth term using the same formula:

[ a_8 = a_1 + (8 - 1)