Find the indicated term of the arithmetic sequence with the given description. The fourteenth term is 2/3 and the ninth term is 1/4. Find the first term and the nth term
First, let's find the common difference of the sequence using the information we have.
Given that the ninth term is 1/4 and the fourteenth term is 2/3, we can set up the following equations:
a + 8d = 1/4 (ninth term)
a + 13d = 2/3 (fourteenth term)
Subtracting the first equation from the second, we get:
5d = 2/3 - 1/4
5d = 8/12 - 3/12
5d = 5/12
d = 1/12
Now that we have the common difference, we can find the first term (a) by substituting back into the equation for the ninth term:
a + 8(1/12) = 1/4
a + 2/3 = 1/4
a = 1/4 - 2/3
a = 3/12 - 8/12
a = -5/12
Therefore, the first term is -5/12 and the nth term can be calculated as:
a + (n-1)d
-5/12 + (n-1)(1/12)
-5/12 + n/12 - 1/12
(n - 6)/12
So, the nth term of the sequence is (n-6)/12.