Find the indicated term of the arithmetic sequence with the given description. The 14th term is 2/3 and the 9th term is 1/4. Find the first term and the nth term.
To find the first term, let's first find the common difference (d) of the arithmetic sequence when given the 9th term and the 14th term:
a9 = a1 + (9-1)d = a1 + 8d = 1/4
a14 = a1 + (14-1)d = a1 + 13d = 2/3
Now we can set up a system of equations to solve for a1 and d:
a1 + 8d = 1/4
a1 + 13d = 2/3
Subtracting the first equation from the second:
5d = 2/3 - 1/4
5d = 8/12 - 3/12
5d = 5/12
d = 1/12
Now we can substitute d back into the first equation to solve for a1:
a1 + 8(1/12) = 1/4
a1 + 2/3 = 1/4
a1 = 1/4 - 8/12
a1 = 3/12 - 8/12
a1 = -5/12
Therefore, the first term of the arithmetic sequence is -5/12.
To find the nth term, we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n-1)d
Substitute the values we found for a1 and d into the formula:
an = -5/12 + (n-1)(1/12)
an = -5/12 + (n/12) - 1/12
an = (-5 + n - 1)/12
an = (n - 6)/12
Therefore, the nth term of the arithmetic sequence is (n - 6)/12.