a Calculate f(3) using the known values of f(1) and f(2).
B
b Calculate the first and second differences.
C
c Use the conclusion from Part B.
A
a 3, 10, 21, 36, 55
B
b See solution.
C
c f(n)=2n^2+n
Practice makes perfect
a Let's analyze the given recursive sequence.
f(1)=3, f(2)=10
f(n)=4+2f(n-1)-f(n-2)
We are asked to write the first five terms of f(n). Let's do it!
n
f(n-2)
f(n-1)
4+2f(n-1)-f(n-2)
f(n)=4+2f(n-1)-f(n-2)
1
-
-
-
f(1)=3
2
-
-
-
f(2)=10
3
f(1)= 3
f(2)= 10
4+2( 10)- 3= 21
f(3)= 21
4
f(2)= 10
f(3)= 21
4+2( 21)- 10= 36
f(4)= 36
5
f(3)= 21
f(4)= 36
4+2( 36)- 21= 55
f(5)= 55
b
We will use patterns between consecutive data pairs to determine what type of function best models the given values from Part A. The differences of consecutive f-values are called first differences. The differences of consecutive first differences are called second differences.
Linear Function: The first differences are constant.
Quadratic Function: The second differences are constant.
Exponential Function: Consecutive y-values have a common ratio.
Remember that in all cases the differences of consecutive n-values need to be constant! Let's analyze the table of values and compute the first differences.
Since the first differences are not constant, the sequence cannot be modeled by a linear function. Next we will check the second differences.
Since the second differences are constant, the sequence can be modeled by a quadratic function.
c From Part B we know that the sequence can be modeled by a quadratic function, f(n)=an^2+bn+c, for some values of a, b, and c. To determine the values of a, b, and c, let's substitute 1, 2, and 3 for n and use that f( 1)= 3, f( 2)= 10, f( 3)= 21.
f(n)=an^2+bn+c
⇓
3=a( 1)^2+b( 1)+c & (I) 10=a( 2)^2+b( 2)+c & (II) 21=a( 3)^2+b( 3)+c & (III)
Let's use the Substitution Method to find a solution to this system.
3=a(1)^2+b(1)+c & (I) 10=a(2)^2+b(2)+c & (II) 21=a(3)^2+b(3)+c & (III)
â–Ľ
Solve by substitution
(I), (II), (III):Calculate power and product
3=a+b+c & (I) 10=4a+2b+c & (II) 21=9a+3b+c & (III)
(I), (II), (III):Rearrange equation
a+b+c=3 & (I) 4a+2b+c=10 & (II) 9a+3b+c=21 & (III)
Finally, an explicit rule for the given sequence is f(n)=2n^2+n.
Checking the Answer
Since we obtain the explicit rule f(n)=2n^2+n based only on a few first terms of the given sequence, we are not sure if this is correct for all values of n. To verify our answer, we should substitute f(n)=2n^2+n into the recursive rule for f(n) and check if it is true.
f(n)=4+2f(n-1)-f(n-2)
First, let's find f(n-1) and f(n-2).
f( n)=2 n^2+ n
⇓
f( n-1)=2( n-1)^2+( n-1)
f( n-2)=2( n-2)^2+( n-2)
Next, we will substitute 2n^2+n for f(n), 2(n-1)^2+(n-1) for f(n-1), and 2(n-2)^2+(n-2) for f(n-2) into the recursive formula.
