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
Resh N. first term is 8 and the 6th term is 243 over 128. i can't get the common ratio. may i ash your help?
2 Answers By Expert Tutors
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Problem:
» What is the 6th term of the geometric sequence if the first term is 6 and the common ratio is 4?
Answer:
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Given:
Solution:
- Since r = 4, just multiply it to the previous term to obtain next term.
Thus, the 6th term of a geometric sequence whose a1 = 6 and r = 4 is 6,144.
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- We can also use formula to get the nth term of a geometric sequence.
Formula:
Solution:
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A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.
We can write a formula for the n th term of a geometric sequence in the form
a n = a r n ,
where r is the common ratio between successive terms.
Example 1:
{ 2 , 6 , 18 , 54 , 162 , 486 , 1458 , ... }
is a geometric sequence where each term is 3 times the previous term.
A formula for the n th term of the sequence is
a n = 2 3 ( 3 ) n
Example 2:
{ 12 , − 6 , 3 , − 3 2 , 3 4 , − 3 8 , 3 16 , ... }
is a geometric series where each term is − 1 2 times the previous term.
A formula for the n th term of this sequence is
a n = 24 ( − 1 2 ) n
Example 3:
{ 1 , 2 , 6 , 24 , 120 , 720 , 5040 , ... }
is not a geometric sequence. The first ratio is 2 1 = 2 , but the second ratio is 6 2 = 3 .
No formula of the form
a n = a r n can be written for this sequence.
See also arithmetic sequences .