The fifth term of an arithmetic sequence is 5.4, and the 12th term is 11.0. Find the nth term.
The second term of a geometric sequence is 28, and the sixth term is 7168. Find the nth term.
The second term of a geometric sequence is 28, and the sixth term is 7168. Find the nth term.
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General formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n-1)d
5.4 = a_5 = a_1 + (5-1)d = a_1 + 4d
11.0 = a_12 = a_1 + (12-1)d = a_1 + 11d
Subtract the first equation from the second:
5.6 = 7d
0.8 = d
Put this in either of the two equations and solve for a_1. You should get a_1 = 2.2
a_n = 2.2 + (n-1)(0.8) = 1.4 + 0.8n
General formula for the nth term of an geometric sequence:
a_n = (a_1)(r^(n-1)) (or a_0 r^n)
28 = a_2 = (a_1)(r^(2-1)) = (a_1)r
7168 = a_6 = (a_1)(r^(6-1)) = (a_1)r^5
Divide the second equation by the first one:
7168/28 = r^(5-1)
256 = r^4
4 = r
Substitute this into either of the equations. You should get a_1 = 7.
a_n = 7*4^(n-1) = (7/4)*(4^n)
a_n = a_1 + (n-1)d
5.4 = a_5 = a_1 + (5-1)d = a_1 + 4d
11.0 = a_12 = a_1 + (12-1)d = a_1 + 11d
Subtract the first equation from the second:
5.6 = 7d
0.8 = d
Put this in either of the two equations and solve for a_1. You should get a_1 = 2.2
a_n = 2.2 + (n-1)(0.8) = 1.4 + 0.8n
General formula for the nth term of an geometric sequence:
a_n = (a_1)(r^(n-1)) (or a_0 r^n)
28 = a_2 = (a_1)(r^(2-1)) = (a_1)r
7168 = a_6 = (a_1)(r^(6-1)) = (a_1)r^5
Divide the second equation by the first one:
7168/28 = r^(5-1)
256 = r^4
4 = r
Substitute this into either of the equations. You should get a_1 = 7.
a_n = 7*4^(n-1) = (7/4)*(4^n)