Question

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.

Answer 1

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.