Recurrence relationships for GP's
The formula for the $n$nth term of a geometric sequence is given by $t_n=ar^{n-1}$tn=arn−1, but there is another way to express the geometric relationship between terms.
It is generally known as a recurrence relationship and for geometric sequences, the recurrence formula is given by:
$t_{n+1}=r\times t_n,t_1=a$tn+1=r×tn,t1=a
The equation states that the $\left(n+1\right)$(n+1)th term is $r$r times the $n$nth term with the first term equal to $a$a.
Thus the second term, $t_2$t2 is $r$r times the first term $t_1$t1, or $ar$ar.
The third term $t_3$t3 is $r$r times $t_2$t2 or $ar^2$ar2.
The fourth term $t_4$t4 is $r$r times $t_3$t3, or $ar^3$ar3, and so on.
Hence, step by step, the sequence is revealed as $a$a, $ar$ar, $ar^2$ar2, $ar^3...$ar3... , $ar^{n-1}$arn−1.
Take for example the recursive relationship given as $t_{n+1}=\frac{t_n}{2}$tn+1=tn2 with $t_1=64$t1=64. From this formula, we see that
$t_2=\frac{t_1}{2}=32$t2=t12=32 and
$t_3=\frac{t_2}{2}=16$t3=t22=16, and so on.
This means that the sequence becomes $64,32,16,8,...$64,32,16,8,... which is clearly geometric with $a=64$a=64 and $r=\frac{1}{2}$r=12.
Consider the recurrence relationship given as $t_{n+1}=3t_n+2$tn+1=3tn+2 with $t_1=5$t1=5.
To test whether or not the relationship is geometric, we can evaluate the first three terms.
$t_1=5$t1=5,
$t_2=3\times5+2=17$t2=3×5+2=17
$t_3=3\times17+2=53$t3=3×17+2=53.
Thus, the sequence begins $5,17,53,...$5,17,53,... and we immediately see that $\frac{53}{17}$5317 is not the same fraction as $\frac{17}{5}$175, and thus the recursive relationship is not geometric. In fact the only way the relationship given by $t_{n+1}=rt_n+k$tn+1=rtn+k is geometric is when the constant term $k$k is zero.
Worked Examples
Question 1
Consider the first-order recurrence relationship defined by $T_n=2T_{n-1},T_1=2$Tn=2Tn−1,T1=2.
Determine the next three terms of the sequence from $T_2$T2 to $T_4$T4.
Write all three terms on the same line, separated by commas.
Plot the first four terms on the graph below.
Loading Graph...Is the sequence generated from this definition arithmetic or geometric?
Arithmetic
AGeometric
BNeither
C
Question 2
The first term of a geometric sequence is $5$5. The third term is $80$80.
Solve for the possible values of the common ratio, $r$r, of this sequence.
State the recursive rule for $T_n$Tn and the initial condition $T_1$T1 that define the sequence with a positive common ratio.
Write both parts of the relationship on the same line, separated by a comma.
State the recursive rule for $T_n$Tn and the initial condition $T_1$T1 that define the sequence with a negative common ratio.
Write both parts of the relationship on the same line, separated by a comma.
Question 3
The average rate of depreciation of the value of a Ferrari is $14%$14% per year. A new Ferrari is bought for $\$90000$$90000.
What is the car worth after $1$1 year?
What is the car worth after $3$3 years?
Write a recursive rule for $V_n$Vn defining the value of the car after $n$n years, and an initial condition $V_0$V0.
Write both parts of the rule on the same line, separated by a comma.