Ideas
Model linear growth and decay
An arithmetic sequence can be used to model linear growth or decay occurs when a quantity increases or decreases by the same amount at regular time intervals.
For any arithmetic sequence with starting value a and constant increase of d, we can express it in either of the following two forms:
Recursive form is a way to express any term in relation to the previous term: V_{n+1}=V_n +d, where V_0=a.
Explicit form is a way to express any term in relation to the term number: V_n= a + (n-1)d, where V_1=a.
If d is positive the sequence with model linear growth and if d is negative it will model linear decay.
Note: If the initial term is set to V_0 rather than V_1, the general rule becomes V_n=a+nd.
Examples
Example 1
The value of a new car decreases every year by \$500. It was originally worth \$45000. Write a recurrence relation to show the depreciation of the value of the car.
For any arithmetic sequence with starting value a and constant increase of d, we can express it in either of the following two forms:
Recursive form is a way to express any term in relation to the previous term: V_{n+1}=V_n +d, where V_0=a.
Explicit form is a way to express any term in relation to the term number: V_n= a + (n-1)d, where V_1=a.
If d is positive the sequence with model linear growth and if d is negative it will model linear decay.
Model geometric growth and decay
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. This models a quantity which increases or decreases by the same percentage rate at regular intervals such as compound interest and reducing balance depreciation.
For any geometric sequence with starting value a and common ratio R, we can express it in either of the following two forms:
Recursive form is a way to express any term in relation to the previous term: V_{n+1}=RV_n, where V_0=a.
Explicit form is a way to express any term in relation to the term number: V_n=aR^{n-1}, where V_1=a.
If R \gt 1 the sequence with model geometric growth and if 0 \geq R \leq it will model geometric decay.
Note: If the initial term is set to V_0 rather than V_1, the general rule becomes V_n=aR^n.
Examples
Example 2
A bouncy ball is dropped onto the ground from a height of 13 metres.
On each bounce, the ball reaches a maximum height of 50\% of its previous maximum height.
Write a recursive rule for a_{n+1}, the height of the ball on the (n+1)th th bounce, in terms of a_n and an initial condition a_0. Write both parts on the same line separated by a comma.
Write a formula for a_n, for the height reached on the nth bounce in terms of n.
How high does the bouncy ball reach after the 4th bounce? Round your answer to two decimal places.
For any geometric sequence with starting value a and common ratio R, we can express it in either of the following two forms:
Recursive form is a way to express any term in relation to the previous term: V_{n+1}=RV_n, where V_0=a.
Explicit form is a way to express any term in relation to the term number: V_n=aR^{n-1}, where V_1=a.
If R \gt 1 the sequence with model geometric growth and if 0 \geq R \leq it will model geometric decay.
Combine linear and geometric growth and decay
Arithmetic and geometric sequences are special forms of a first order linear recurrence relation. The first order in the name refers to the relationship only looking back one term to define the next and linear as we will not raise the previous term to a power other than one. This means we can write each term as a linear combination of the previous term, so the general form is: V_{n+1}=RV_n +d, where R and d are constants. Such a recurrence relation can be used more generally to model situations that include both linear and geometric growth or decay.
V_0 = \text{initial value}, \, V_{n+1} = RV_n +d where R is the geometric componenent and d is the linear component.
Examples
Example 3
Consider the recurrence relation U_{n+1}=0.5U_n +2 and U_0=20.
Find U_1.
Find U_2.
Find U_3.
Complete the table of values.
| n | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| U_n |
Graph the relation.