REVIEW OF GEOMETRIC SEQUENCES
The sequence shown below
2 , 8 , 32 , 128 , ...
has been obtained starting from 2 and multiplying each term by 4. 2 is the first term of the sequence and 4 is the common ratio.
8 = 2 * 4 or 8 / 2 = 4
32 = 8 * 4 or 32 / 8 = 4
128 = 32 * 4 or 128 / 32 = 4 and so on.
The terms in the sequence may also be written as follows
a_{1} = 2
a_{2} = a_{1} * 4 = 2 * 4
a_{3} = a_{2} * 4 = 2 * 4^{2}
a_{4} = a_{3} * 4 = 2 * 4^{3}
The n th term may now be written as
a_{n} = a_{1} * r^{n1}
where a_{1} is the first term of the sequence and r is the common ratio which is equal to 4 in the above example.
The sum of the first n terms of a geometric sequence is given by
s_{n} = a_{1} + a_{2} + a_{3} + ... a_{n} = a_{1} (1  r^{n}) / (1  r)
The sum S of an infinite (n approaches infinity) geometric sequence and when r <1 is given by
S = a_{1} / (1  r)
Problem 1:
Find the terms a_{2}, a_{3}, a_{4} and a_{5} of a geometric sequence if a_{1} = 10 and the common ratio r =  1.
Solution to Problem 1:
 Use the definition of a geometric sequence
a_{2} = a_{1} * r = 10 * (1) =  10
a_{3} = a_{2} * r =  10 * (1) = 10
a_{4} = a_{3} * r = 10 * (1) =  10
a_{5} = a_{4} * r =  10 * (1) = 10
Problem 2:
Find the 10 th term of a geometric sequence if a_{1} = 45 and the common ration r = 0.2.
Solution to Problem 2:
 Use the formula a_{n} = a_{1} * r^{n1} that gives the n th term to find a_{10} as follows
a_{10} = a_{1} * r^{n1}
= 45 * 0.2^{9} = 2.304 * 10^{5}
Problem 3:
Find a_{20} of a geometric sequence if the first few terms of the sequence are given by
1/2 , 1/4 , 1/8 , 1 / 16 , ...
Solution to Problem 3:
 We first use the first few terms to find the common ratio
r = a_{2} / a_{1} = (1/4) / (1/2) = 1/2
r = a_{3} / a_{2} = (1/8) / (1/4) = 1/2
r = a_{4} / a_{3} = (1/16) / (1/8) = 1/2
 The common ration r = 1/2. We now use the formula a_{n} = a_{1} * r^{n1} for the n th term to find a_{20} as follows.
a_{20} = a_{1} * r^{201}
= (1/2) * (1/2)^{201} = 1 / (20^{20})
Problem 4:
Given the terms a_{10} = 3 / 512 and a_{15} = 3 / 16384 of a geometric sequence, find the exact value of the term a_{30} of the sequence.
Solution to Problem 4:
 We first use the formula for the n th term to write a_{10} and a_{15} as follows
a_{10} = a_{1} * r^{101} = 3 / 512
a_{15} = a_{1} * r^{151} = 3 / 16384
 We now divide the terms a_{10} and a_{15} to write
a_{15} / a_{10} = (a_{1} * r^{14} / a_{1} * r^{9}) = (3 / 16384) / (3 / 512)
 Solve for r to obtain.
r^{5} = 1 / 32 which gives r = 1/2
 We now use a_{10} to find a_{1} as follows.
a_{10} = 3 / 512 = a_{1}(1/2)^{9}
 Solve for a_{1} to obtain.
a_{1} = 3
 We now use the formula for the n th term to find a_{30} as follows.
a_{30} = 3(1/2)^{29} = 3 / 536870912
Problem 5:
Find the sum
Solution to Problem 5:
 We first rewrite the sum S as follows
S = 1 + 3 + 9 + 27 + 81 + 243 = 364
 Another method is to first note that the terms making the sum are those of an arithmetic sequence with a_{1} = 1 and r = 3 using the formula s_{n} = a_{1} (1  r^{n}) / (1  r) with n = 6.
s_{6} = 1 (1  3^{6}) / (1  3) = 364
Problem 6:
Find the sum
10 
S = ∑ 8*(1/4)^{i1} 
i=1 
Solution to Problem 6:
 An examination of the terms included in the sum are
8 , 8*((1/4)^{1} , 8*((1/4)^{2} , ... , 8*((1/4)^{9}
 These are the terms of a geometric sequence with a_{1} = 8 and r = 1/4 and therefore we can use the formula for the sum of the terms of a geometric sequence
s_{10} = a_{1} (1  r^{n}) / (1  r)
= 8 * (1  (1/4)^{10}) / (1  1/4) = 10.67 (rounded to 2 decimal places)
Problem 7:
Write the rational number 5.31313131... as the ratio of two integers.
Solution to Problem 7:
 We first write the given rational number as an infinite sum as follows
5.313131... = 5 + 0.31 + 0.0031 + 0.000031 + ....
 The terms making 0.31 + 0.0031 + 0.000031 ... are those of a geometric sequence with a_{1} = 0.31 and r = 0.01. Hence the use of the formula for an infinite sum of a geometric sequence
S = a_{1} / (1  r) = 0.31 / (1  0.01) = 0.31 / 0.99 = 31 / 99
 We now write 5.313131... as follows
5.313131... = 5 + 31/99 = 526 / 99
Exercises:
Answer the following questions related to geometric sequences:
a) Find a_{20} given that a_{3} = 1/2 and a_{5} = 8
b) Find a_{30} given that the first few terms of a geometric sequence are given by 2 , 1 , 1/2 , 1/4 ...
c) Find r given that a_{1} = 10 and a_{20} = 10^{18}
d) write the rational number 0.9717171... as a ratio of two positive integers.
Solutions to Above Exercises:
a) a_{20} = 2^{18}
b) a_{30} = 1 / 2^{28}
c) r = 0.1
d) 0.9717171... = 481/495
More math problems with detailed solutions in this site.
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