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
Answer: 1. 320 2. 4 3. 16/243 Step-by-step explanation: use the formula a sub n is equal to a sub 1 r raise to n minus 1 1.20 = a sub 1 • r raise to 5-1 20 = a sub 1 • (½) ⁴ 20 = a sub 1 • 1/16 divide both sides by 1/16 320= a sub 1 / a sub 1 = 320 2.972 = a sub 1 • r raise to 6-1 972 = a sub 1 • (3) raise to 5 972 = a sub 1 • 243 divide both sides by 243 4 = a sub 1 / a sub 1 = 4 3.1/972 = a sub 1 • r raise to 4-1 1/972 = a sub 1 • (¼)³ 1/972 = a sub 1 • 1/64 divide both sides by 1/64 64/972 = a sub 1 16/243 = a sub 1 / a sub 1 = 16/243 |