6.3 Sequences Type II: Geometric Sequences
Sequences where the ratio of consecutive terms is the same.
Example: 5, 25, 125, 625, ....
Formula:
tn = ar^(n-1)
Example One: Find t5 for the geometric sequence 3, 6, 12, 24,...
Given a = 3 and r = 2
t5 = (3) * (2) ^ (5-1)
t5 = 48
Therefore, the 5th term is 48.
Other questions may ask of you to determine the number of terms in a geometric sequence. To do so, just sub in whatever values are given.
Example Two: In a geometric sequence of real numbers, t5 = 3750 and t7 = 93750. Find the first two terms.
Solution:
t7= 1000 ---> ar6 = 93750
t5= 40 ---> ar4 = 3750
ar6/ar4 = 93750/3750
ar6/ar4 = 25
r = _+5
This indicates that either r = 5 or r = -5.
(i) a(5)^4 = 3750
625 a = 3750
a = 6
Therefore t1 = 6, t2 = 30...
(ii) r = - 5
a(-5)^4 = 3750
a =6
Therefore, t1 = 6, t2 = -15...
- A geometric mean is a term between two other terms in a geometric progression.
Some questions may ask of you to find geometric means between two numbers. To do so, sub in the values to find "r". Determine the correct "r" (that is, a real number r) and calculate the sequence.
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