Recurrence Relations : Lines and Planes; Savings and Interest; n-Digit Ternary Sequences

1) Find and solve a recurrence relation for the number of n-digit ternary sequences with no consecutive digits being equal.

2) Find and solve a recurrence relation for the number of infinite regions formed by n infinite lines drawn in the plane so that each pair of lines intersects at a different point.

3) Find and solve a recurrence relation for the number of different regions formed when n mutually intersecting planes are drawn in three-dimensional space such that no four planes intersect at a common point and no two planes have parallel intersection lines in a third plane. (Hint: reduce to a two-dimensional problem.)

4) Suppose a savings account earns 5 percent a year. Initially there is $1000 in the account, and in year k, $10k are withdrawn. How much money is in the account at the end of n years if:
A) Annual withdrawal is at year's end?
B) Withdrawal is at start of year?

5) Find and solve a recurrence relation for the number of n-digit ternary sequences in which no 1 appears to the right of any 2.

Recurrence Relations and Lines and Planes; Savings and Interest; n-Digit Ternary Sequences are investigated. The solution is detailed and well presented.

1. Find a functional equation and solve it for sequence of generating functions whose coefficients satisfy (assume and =1):
1.
2. Find a recurrence relation and associated generating function for the number of n-digitternary sequence that have the pattern "012" occurring for the first time at end of the sequence.

Show that the Fibonacci numbers satisfy the recurrence relation f_n = 5f_n-4 + 3f_n-5 for n = 5, 6, 7,..., together with the initial f_0 = 0, f_1 = 1, f_2 = 1, f_3 = 2, and f_4 = 3.
See the attachment for the full question.

Find the solution to
an = 2an-1 + an-2 -2an-3
for n= 3,4,5,..., with a0 = 3, a1=6, and a2=0.
Find the solution to
an= 5an-2 - 4an-4
with a0 = 3, a1 = 2, a2 = 6, and a3 = 8.
(I have attached the full problem below.)

Please help with the following discrete math involving recurrencerelations, compound interest, polynomials, number of combinations and iteration.
14. Individual membership fees at the evergreen tennis club were $50 in 1970 and have increased by $2 per year since then. Write a recurrence relation and initial conditions for

Zebra mussels are fresh water mollusks that attack underwater structures. Suppose that the volume of mussels in a confined area grows at a rate of 0.2% per day
(1) If there are now 10 cubic feet of mussels in a lock on the Illinois River at Peoria, Illinois, develop a recurrence relation and initial conditions that represent

See attached file.
Here are some recurrencerelations that come up in the analysis of Quicksort and or Select. For each recurrence, write one sentence explaining how it relates to Quicksort or Select and give its asymptotic growth rate in notation.
A. T(n) = T(n/10) + T(9n/10) + n
B. T(n) = T(n-1) + n
C. T(n) = T(n

6. A string that contains only 0s 1s and 2s is called a ternary string
a) find a recurrence relation for the number of ternary strings that contain two consecutive 0s
b) what are the initial conditions
c) how many ternary strings of length six contain two consecutive 0s
The next 3 problems deal with a variation of the

1. Find the sequence for the recursive formula:
S_n = -s_n-1 + 9, s_0 = -3
(see the attachment for the full question)
2. True of False a_n = 2 is a solution to the recurrence relation a_n = 2a_n-1 - a_n-2 with initial conditions a_0 = 2 and a_1 = 2.
a. True
b. False
3. Find a solution to the recurrence relation:
a_