A sequence in which each term increases or decreases from the last by a constant factor is called a geometric sequence. We refer to the constant factor the terms are changing by as the common ratio, which will result from dividing any two successive terms $\left(\frac{t_{n+1}}{t_n}\right)$(tn+1tn).
We denote the first term in the sequence by the letter $a$a and the common ratio by the letter $r$r. For example, the sequence $4,8,16,32\dots$4,8,16,32… is geometric with $a=4$a=4 and $r=2$r=2. The sequence $100,-50,25,-12.5,\dots$100,−50,25,−12.5,… is geometric with $a=100$a=100 and $r=\frac{-1}{2}$r=−12.
To describe the rule in words we say "next term is $r$r multiplied by previous term". Therefore we can write any geometric sequence as the recurrence relation:
$t_n=rt_{n-1},t_1=a$tn=rtn−1,t1=a or alternatively $t_{n+1}=rt_n$tn+1=rtn, where $t_1=a$t1=a
Note that it is also possible to define the initial term as $t_0$t0, this is particularly useful in financial applications of sequences.
We can also find an explicit formula in terms of $a$a and $r$r, this is useful for finding the $n$nth term without listing the sequence.
Let's look at a table of a concrete example to see the pattern for the explicit formula. For the sequence $5,10,20,40,\ldots$5,10,20,40,…, we have a starting term of $5$5 and a common ratio of $2$2, that is $a=5$a=5 and $r=2$r=2. A table of the sequence is show below:
$n$n | $t_n$tn | Pattern |
---|---|---|
$1$1 | $5$5 | $5\times2^0$5×20 |
$2$2 | $10$10 | $5\times2^1$5×21 |
$3$3 | $20$20 | $5\times2^2$5×22 |
$4$4 | $40$40 | $5\times2^3$5×23 |
... | ||
$n$n | $t_n$tn | $5\times2^{n-1}$5×2n−1 |
The pattern starts to become clear and we could guess that the tenth term is $t_{10}=5\times2^9$t10=5×29 and the one-hundredth term is $t_{100}=5\times2^{99}$t100=5×299. And following the pattern, the explicit formula for the $n$nth term is $t_n=5\times2^{n-1}$tn=5×2n−1.
For any geometric progression with starting value $a$a and common ratio $r$r has the terms given by: $a,ar,ar^2,ar^3,...$a,ar,ar2,ar3,... We see a similar pattern to our previous table and can write down the formula for the $n$nth term:
$t_n=ar^{n-1}$tn=arn−1
For any geometric sequence with starting value $a$a and common ratio $r$r, we can express it in either of the following two forms:
$t_n=rt_{n-1}$tn=rtn−1, where $t_1=a$t1=a or alternatively $t_{n+1}=rt_n$tn+1=rtn, where $t_1=a$t1=a
$t_n=ar^{n-1}$tn=arn−1
For the sequence $15,30,60,120...$15,30,60,120..., find an explicit rule for the $n$nth term and hence, find the $8$8th term.
Think: Check that the sequence is geometric, is the next term made by multiplying the previous term by a constant factor? Then write down the starting value $a$a and common ratio $r$r and substitute these into the general form: $t_n=ar^{n-1}$tn=arn−1
Do: Dividing the second term by the first we get, $\frac{t_2}{t_1}=\frac{2}{1}$t2t1=21. Checking the ratio between the successive pairs we also get $\frac{2}{1}$21. So we have a geometric sequence with: $a=15$a=15 and $r=2$r=2. The general formula for this sequence is: $t_n=15\left(2\right)^{n-1}$tn=15(2)n−1.
Hence, the $8$8th term is: $t_8=15\left(2\right)^7=15\times128=1920$t8=15(2)7=15×128=1920.
If a geometric sequence has $t_3=12$t3=12 and $t_6=96$t6=96, find the recurrence relation for the sequence.
Think: To find the recurrence relation we need the starting value and common ratio. There are two methods we can use to do this.
Method 1:
Think: Determine how many times $t_3$t3 is multiplied by $r$r to make $t_6$t6
Do:
$t_3\times r\times r\times r$t3×r×r×r | $=$= | $t_6$t6 |
$12\times r^3$12×r3 | $=$= | $96$96 |
$r^3$r3 | $=$= | $8$8 |
$\therefore r$∴r | $=$= | $2$2 |
To find $a$a think: if we divide $t_3$t3 by $r$r we get $t_2$t2. Then divide $t_2$t2 by $r$r to find $t_1$t1 or $a$a.
Therefore $a=12\div2\div2=3.$a=12÷2÷2=3.
Think: We have a GP with $a=3$a=3 and $r=2$r=2. In words we can say "next term is two times previous term, with first term of $3$3".
Do: Write - The recurrence relation for this sequence is:
$t_n=2t_{n-1},t_1=3$tn=2tn−1,t1=3
Method 2:
Think: As we have two terms we can set up two equations in terms of $a$a and $r$r using $t_n=ar^{n-1}$tn=arn−1.
Do:
$t_3$t3: | $ar^2=12$ar2=12 | $.....\left(1\right)$.....(1) |
and
$t_6$t6: | $ar^5=96$ar5=96 | $.....\left(2\right)$.....(2) |
If we now divide equation $\left(2\right)$(2) by equation $\left(1\right)$(1), we obtain the following:
$\frac{ar^5}{ar^2}$ar5ar2 | $=$= | $\frac{96}{12}$9612 |
$r^3$r3 | $=$= | $8$8 |
$\therefore r$∴r | $=$= | $2$2 |
With the common ratio found to be $2$2, then we know that, using equation $\left(1\right)$(1) $a\times2^2=12$a×22=12 and so $a$a is $3$3. The recurrence relation for this sequence is given by:
$t_n=2t_{n-1},t_1=3$tn=2tn−1,t1=3
Study the pattern for the following geometric sequence, and write down the next two terms.
$3$3, $15$15, $75$75, $\editable{}$, $\editable{}$
Consider the first four terms in this geometric sequence: $-8$−8, $-16$−16, $-32$−32, $-64$−64
If $T_n$Tn is the $n$nth term, evaluate $\frac{T_2}{T_1}$T2T1.
Evaluate $\frac{T_3}{T_2}$T3T2
Evaluate $\frac{T_4}{T_3}$T4T3
Hence find the value of $T_5$T5.
In a geometric progression, $T_4=54$T4=54 and $T_6=486$T6=486.
Solve for $r$r, the common ratio in the sequence. Write both solutions on the same line separated by a comma.
For the case where $r=3$r=3, solve for $a$a, the first term in the progression.
Consider the sequence in which the first term is positive. Find an expression for $T_n$Tn, the general $n$nth term of this sequence.
When given a formula for the $n$nth term we can generate a table of values for the sequence. For example in the sequence given by the formula $t_n=12\times\left(1.5\right)^{n-1}$tn=12×(1.5)n−1, by substituting for $n$n appropriately and using a calculator, we can generate the following table of the first $6$6 terms of the sequence:
$n$n | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 |
---|---|---|---|---|---|---|
$t_n$tn | $12$12 | $18$18 | $27$27 | $40.5$40.5 | $60.75$60.75 | $91.125$91.125 |
Perhaps more interesting though is the different types of graphs that geometric sequences correspond to. The graphs are not linear like arithmetic progressions, except for the trivial case of $r=1$r=1. The path of points plotted from a geometric sequence follow an exponential curve for positive values of $r$r:
Sequences of the form $t_n=ar^{n-1}$tn=arn−1 are exponential graphs, and where $a>0$a>0 they will follow:
What happens when $r$r is negative? The values of successive terms flip their sign so that the graph is depicted as either a growing($r<-1$r<−1) or diminishing($-1
For the geometric progression with starting value $12$12 and ratio $r=-1.5$r=−1.5, create a table and plot a graph of the sequence.
Think: This is the same as the example in the previous table but the ratio is now negative. The $n$nth term is given by $t_n=12\times\left(-1.5\right)^{n-1}$tn=12×(−1.5)n−1, the table will be the same but the sign of the terms will alternate.
Do: The new table becomes:
$n$n | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 |
---|---|---|---|---|---|---|
$t_n$tn | $12$12 | $-18$−18 | $27$27 | $-40.5$−40.5 | $60.75$60.75 | $-91.125$−91.125 |
Checking, for $n=1$n=1, we have $t_1=12\times\left(-1.5\right)^{1-1}=12$t1=12×(−1.5)1−1=12 and for $n=2$n=2 we have $t_2=12\times\left(-1.5\right)^{2-1}=-18$t2=12×(−1.5)2−1=−18, continuing on even numbered terms become negative and odd numbered terms become positive.
Here is a graph of the two geometric sequences depicted in both tables. Note that the odd terms of the zig-zag graph coincide with the terms of the first geometric progression.
Try adjusting the values of $a$a and $r$r in the applet below to observe the effect on the plotted points.
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The $n$nth term of a geometric progression is given by the equation $T_n=2\times3^{n-1}$Tn=2×3n−1.
Complete the table of values:
$n$n | $1$1 | $2$2 | $3$3 | $4$4 | $10$10 |
---|---|---|---|---|---|
$T_n$Tn | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
What is the common ratio between consecutive terms?
Plot the points in the table that correspond to $n=1$n=1, $n=2$n=2, $n=3$n=3 and $n=4$n=4.
If the plots on the graph were joined they would form:
a straight line
a curved line
Consider the following graph of the first $4$4 terms of a sequence.
Write a recursive rule for $T_n$Tn in terms of $T_{n-1}$Tn−1, including the initial term $T_1$T1.
Enter both parts on the same line, separated by a comma.
The given table of values represents terms in a geometric sequence.
$n$n | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|
$T_n$Tn | $7$7 | $-21$−21 | $63$63 | $-189$−189 |
Identify $r$r, the common ratio between consecutive terms.
Write a simplified expression for the general $n$nth term of the sequence, $T_n$Tn.
Find the $12$12th term of the sequence.