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.
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.