Find a formula for the nth term of the recurrence K(n) = 13.5K(n – 1) – 2.7 with K(0) = 5.
To find a formula for the nth term of the recurrence K(n) = 13.5K(n – 1) – 2.7 with K(0) = 5, we can use a technique called iteration.
Let's compute the first few terms of the sequence to identify a pattern:
K(0) = 5
K(1) = 13.5K(0) - 2.7 = 13.5(5) - 2.7 = 67.5 - 2.7 = 64.8
K(2) = 13.5K(1) - 2.7 = 13.5(64.8) - 2.7 = 874.8 - 2.7 = 872.1
K(3) = 13.5K(2) - 2.7 = 13.5(872.1) - 2.7 = 11761.35 - 2.7 = 11758.65
...
Based on the calculated terms, it seems that each term is obtained by multiplying the previous term by 13.5 and subtracting 2.7. We can represent this relationship with the formula K(n) = 13.5K(n – 1) – 2.7.
Now, let's use iteration to find a general formula for the sequence.
Step 1: Express K(n) in terms of K(n-1):
K(n) = 13.5K(n – 1) – 2.7
Step 2: Substitute the expression from Step 1 into itself for K(n – 1):
K(n) = 13.5(13.5K(n – 2) – 2.7) – 2.7
Expanding the equation:
K(n) = (13.5^2)K(n – 2) – 13.5(2.7) – 2.7
Step 3: Repeat this process until you reach K(0):
K(n) = (13.5^3)K(n – 3) – 13.5(2.7^2) – 13.5(2.7) – 2.7
...
K(n) = (13.5^n)K(0) – 13.5(2.7^(n-1)) – 13.5(2.7^(n-2)) – ... – 13.5(2.7) – 2.7
Since K(0) = 5, we can substitute that into the formula:
K(n) = (13.5^n)(5) – 13.5(2.7^(n-1)) – 13.5(2.7^(n-2)) – ... – 13.5(2.7) – 2.7
Therefore, the formula for the nth term of the given recurrence relation K(n) = 13.5K(n – 1) – 2.7 with K(0) = 5 is:
K(n) = (13.5^n)(5) – 13.5(2.7^(n-1)) – 13.5(2.7^(n-2)) – ... – 13.5(2.7) – 2.7