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.

Answer 1

To find the first term (a) and the nth term of the arithmetic sequence, we can first find the common difference (d) by using the formula:

d = (ninth term - first term) / 8
d = (1/4 - a) / 8

We are also given that the fourteenth term is 2/3, so we can use the formula for the nth term of an arithmetic sequence to find the value of n:

nth term = a + (n - 1)d
2/3 = a + (14 - 1)d

Now we can substitute the value of d into the equation for the common difference and solve for a:

(1/4 - a) / 8 = d
(1/4 - a) / 8 = (2/3 - a) / 13
13(1/4 - a) = 8(2/3 - a)
13/4 - 13a = 16/3 - 8a
39 - 52a = 64 - 32a
14a = 25
a = 25/14

Now we can substitute the value of a into the equation for the nth term and solve for n:

2/3 = 25/14 + 13(14 - 1)d
2/3 = 25/14 + 13(13)d
2/3 = 25/14 + 169d
2 * 14 = 25 * 3 + 169(42)d
28 = 75 + 169(42)d
28 = 75 + 7118d
28 = 75 + 7118d
-47 = 7118d
-47/7118 = d

Therefore, the first term of the arithmetic sequence is 25/14 and the nth term can be represented by the formula (-47/7118)n + 25/14.