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
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