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Given a geometric sequence with the first term a 1 and the common ratio r , the n th (or general) term is given by
a n = a 1 ⋅ r n − 1 .
Example 1:
Find the 6 th term in the geometric sequence 3 , 12 , 48 , ... .
a 1 = 3 , r = 12 3 = 4 a 6 = 3 ⋅ 4 6 − 1 = 3 ⋅ 4 5 = 3072
Example 2:
Find the 7 th term for the geometric sequence in which a 2 = 24 and a 5 = 3 .
Substitute 24 for a 2 and 3 for a 5 in the formula
a n = a 1 ⋅ r n − 1 .
a 2 = a 1 ⋅ r 2 − 1 → 24 = a 1 r a 5 = a 1 ⋅ r 5 − 1 → 3 = a 1 r 4
Solve the firstequation for a 1 : a 1 = 24 r
Substitute this expression for a 1 in the second equation and solve for r .
3 = 24 r ⋅ r 4 3 = 24 r 3 1 8 = r 3 so r = 1 2
Substitute for r in the first equation and solve for a 1 .
24 = a 1 ( 1 2 ) 48 = a 1
Now use the formula to find a 7 .
a 7 = 48 ( 1 2 ) 7 − 1 = 48 ⋅ 1 64 = 3 4
See also: sigma notation of a series and n th term of a arithmetic sequence
Eilidh D. What is the 7th term of the geometric sequence where a1 = -4,096 and a4 = 64?
2 Answers By Expert Tutors
The equation for a geometric sequence is: y = abx ; a is the zero term; b is the common ratio; 64 = -4096 b3 ; solve for b ---------------------------
Kenneth S. answered • 08/29/16
Expert Help in Algebra/Trig/(Pre)calculus to Guarantee Success in 2018
This may be easy, if we recognize that the signs must alternate, and 4096 = 212 and 64 = 26.
To get from first to fourth term, we must do three multiplications by r, so the difference in exponents (12-6) means that r must be -1/22
So a2 = -4096•(-¼) = 1024, a3 = 1024•(-¼) = -256, and one more application of r gives you, correctly, the stated a4.
Now you know how to proceed until you get the 7th term.