Find a recurrence relation and initial condition to model the IRA investor in Ex

Question

Find a recurrence relation and initial condition to model the IRA investor in Example 4 if, instead of depositing $2000 at the beginning of every year, he or she deposits that amount at the end of the year (so that it does not earn interest for the year). Suppose that Fibonacci's rabbits in Example 5 take 2 month to mature instead of 1 month. Write down the recurrence relation and initial condition to model the growth of the rabbit population. Find the number of rabbits present at the end of 6 months. Find a recurrence relation and initial conditions for the sequence {a_m} if a_m is the number of bit strings of length n that do not contain three consecutive 0's. Find a recurrence relation and initial conditions for the sequence {a_m} if a_m is the number of bit strings of length n that contain three consecutive 0's. Find a recurrence relation and initial conditions for the number of sequence over the alphabet {a, b, c} that do not contain three consecutive a's. Find a recurrence relation and initial conditions for the number of sequence over the alphabet {a, b, c} that contain three consecutive a's. A child takes either big steps or little steps. The big step cover 20 inches, and the little step cover 10 inches. Let a_n be the number of ways there are for the child to walk 10n inches. Write down the recurrence relation and initial condition for{a_n}. Determine the number of ways for the child to walk 10 feet. Find an explicit formula for the number of partitions of the positive integers k into parts no larger than 2. Let q(n) be the number of partitions of a set with n elements into sets with at most four elements each. Write down a recurrence relation for q, and give the appropriate initial conditions. Compute q(6). Let r(n) be the number of partitions of a set with n elements into sets with at most four elements each. Write down a recurrence relation for r, and give the appropriate initial conditions. Compute r(6).

Explanation / Answer

12) Let S be the set of all such strings.

First define the set inductively but in such a way as to avoid counting the same string twice:

Basic step: 000 is in S

Inductive step(1): if w is in S and u and v are in {0,1} then uwv is in S.

The above definition is adequate to define S but not for counting.

Inductive step(2): if w is in S and u is in {0,1} then

1w

01w

001w

000u are in S.

This yields the recurrence an=an-1+an-2+an-3+2n-3

with initial conditions: a3=1, a4=3, a5=8

- 01000, since 01w is in S

- 00000, 00001, 00010, 00011 since wu is in S

Check:

a6 = a5+a4+a3+23 = 8+3+1+8=20

a5:: 111000, 110000, 110001, 101000, 100000, 100001, 100010, 100011 = 8

a4:: 011000, 010000, 010001 = 3

a3:: 001000 = 1

23: 000000, 000001, 000010, 000011, 000100, 000101, 000110, 000111 = 8

Every string is present and nothing is counted twice.

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