Given a geometric sequence with the first term a 1 and the common ratio r , the n th (or general) term is given by
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
Problem:» What is the 6th term of the geometric sequence if the first term is 6 and the common ratio is 4? Answer:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 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. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - We can also use formula to get the nth term of a geometric sequence. Formula:Solution:_______________∞_______________
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 . |