Arithmetic, Geometric, and Fibonacci Sequences

Part I: Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant.

Problem 1

Consider the arithmetic sequence: 5, 9, 13, 17, ...

a) Find the common difference of this sequence.

b) Determine the explicit formula for the nth term.

c) Calculate the 15th term of the sequence.

d) Find the sum of the first 20 terms of this sequence.

Problem 2

For the arithmetic sequence where a = -3 and d = 4:

a) Write the first five terms of this sequence.

b) Find the value of a (the 25th term).

c) Determine the smallest positive integer n such that a > 100.

d) Calculate the sum of the first 30 terms of this sequence.

Problem 3

In an arithmetic sequence, a = 13 and a = 37.

a) Find the first term a and the common difference d.

b) Write the explicit formula for the nth term.

c) Calculate the sum of the first 15 terms.

d) If the sum of the first n terms is 1225, find the value of n.

Part II: Geometric Sequences

A geometric sequence is a sequence where the ratio between consecutive terms is constant.

Problem 4

Consider the geometric sequence: 3, 6, 12, 24, ...

a) Find the common ratio of this sequence.

b) Determine the explicit formula for the nth term.

c) Calculate the 8th term of the sequence.

d) Find the sum of the first 10 terms of this sequence.

Problem 5

For the geometric sequence where a = 5 and r = 1/2:

a) Write the first six terms of this sequence.

b) Find the value of a (the 9th term).

c) Calculate the sum of the first 12 terms.

d) Determine the sum of all terms of this infinite geometric sequence.

Problem 6

In a geometric sequence, a = 16 and a = 128.

a) Find the first term a and the common ratio r.

b) Write the explicit formula for the nth term.

c) Calculate a (the 10th term).

d) Find the sum of the first 8 terms of this sequence.

Part III: Fibonacci Sequence

The Fibonacci sequence is defined by the recurrence relation: F = 1, F = 1, and F = F + F for n > 2. This means each term (after the first two) is the sum of the two preceding terms.

Problem 7

Consider the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, ...

a) Calculate F, F, and F (the 10th, 11th, and 12th terms).

b) Find the sum of the first 15 Fibonacci numbers.

c) Show that for any n 1, F + F + ... + F = F - 1.

Problem 8

Prove by mathematical induction that for n 1, F + F + ... + F = F F.

Problem 9

The ratio of consecutive Fibonacci numbers approaches the Golden Ratio = (1+5)/2 1.618...

a) Calculate the ratio F/F.

b) Calculate the ratio F/F.

c) How many terms do we need to consider before the ratio of consecutive terms differs from by less than 0.001?

Problem 10

Investigate the relationship between the Fibonacci sequence and Pascal's Triangle:

a) Write out the first 6 rows of Pascal's Triangle (starting with row 0):

Row 0: 1

Row 1: 1 1

Row 2: 1 2 1

Row 3: 1 3 3 1

Row 4: 1 4 6 4 1

Row 5: 1 5 1010 5 1

b) Calculate the sums of elements along the "shallow diagonals" of Pascal's Triangle as illustrated below:

First sum: The element at (0,0) = 1

Second sum: The element at (1,0) = 1

Third sum: The elements at (2,0) + (1,1) = 1 + 1 = 2

Fourth sum: The elements at (3,0) + (2,1) = 1 + 2 = 3

Fifth sum: The elements at (4,0) + (3,1) + (2,2) = 1 + 3 + 1 = 5

Sixth sum: The elements at (5,0) + (4,1) + (3,2) = 1 + 4 + 3 = 8

Where (row, column) indicates the position in Pascal's Triangle (both starting from 0).

Compare these sums to the Fibonacci sequence. Continue this pattern for the next two sums.

c) Explain why this pattern produces the Fibonacci sequence. (Hint: Consider the recurrence relation for generating Pascal's Triangle and how elements along these diagonals relate to each other.)

Problem 11

In the Fibonacci sequence, prove that:

a) For any n 1, F F - F = (-1)

b) For any n, m 1, F = F F + F F

Problem 12

The closed-form expression for the nth Fibonacci number is known as Binet's formula: F = ( - (1-))/5, where = (1+5)/2

a) Use this formula to calculate F and verify your answer by using the recursive definition.

b) Why does this formula always give an integer despite involving irrational numbers and square roots?

c) Use Binet's formula to find F without calculating all previous terms.

Part IV: Mixed Problems

Problem 13

A ball is dropped from a height of 10 meters. Each time it hits the ground, it bounces back to 4/5 of its previous height.

a) Find the height of the ball after the 5th bounce.

b) Calculate the total distance the ball travels before coming to rest (sum of all up and down movements).

c) What percentage of the original height will the ball reach after the 8th bounce?

Problem 14

The sum of the first n terms of an arithmetic sequence is given by S = 2n + 3n.

a) Find the first term a and the common difference d.

b) Calculate the 15th term of this sequence.

c) Find the sum of terms from the 10th to the 20th term (inclusive).

Problem 15

For the sequence defined by a = 32 - 5:

a) Determine if this is an arithmetic sequence, a geometric sequence, or neither. Justify your answer.

b) Find the first four terms of this sequence.

c) If this sequence represents the amount (in thousands of dollars) that a company earns in the nth month of operation, what is the total earning in the first year?

Problem 16

A deposit of $5,000 is made in an account that pays 6% annual interest compounded monthly.

a) Write a sequence that represents the balance after each month for the first year.

b) Find the balance after 3 years.

c) How many months will it take for the initial deposit to double?

Problem 17

A population of rabbits follows the Fibonacci pattern of growth. Starting with one pair of baby rabbits, each pair matures in one month and then produces a new pair of baby rabbits each month thereafter.

a) How many pairs of rabbits will there be after 8 months?

b) Of these rabbit pairs, how many are mature (more than 1 month old)?

c) If each pair of rabbits requires 2 square meters of space, how much space will be needed to house all rabbits after 1 year?

Problem 18

Consider a sequence formed by combining properties of arithmetic and Fibonacci sequences: a = F + nd, where F is the nth Fibonacci number and d is a constant.

a) If d = 3, find the first 8 terms of this sequence.

b) Is this sequence arithmetic, geometric, or neither? Justify your answer.

c) Find a formula for the sum of the first n terms of this sequence.

Part V: Proofs and Derivations

Problem 19

Prove that the sum of the first n terms of an arithmetic sequence with first term a and common difference d is: S = n/2[2a + (n-1)d]

Problem 20

Prove that the sum of the first n terms of a geometric sequence with first term a and common ratio r (where r 1) is: S = a(1-r)/(1-r)

Problem 21

For an arithmetic sequence with common difference d, prove that: a + a = a + a where p is any integer such that m-p 1

Problem 22

For a geometric sequence with first term a and common ratio r, prove that the product of the first n terms is: P = ar^(n(n-1)/2)

Problem 23

Prove the following Fibonacci identities:

a) F + F + F + ... + F = F

b) F + F + F + ... + F = F - 1

Problem 24

The Lucas numbers are defined similarly to Fibonacci numbers but with different starting values: L = 1, L = 3, and L = L + L for n > 2.

a) Calculate the first 10 Lucas numbers.

b) Prove that for all n 1, L = F + F.

c) Find a relationship between L and Binet's formula.

Problem 25

Examine the ratio of consecutive Lucas numbers (L/L) as n approaches infinity:

a) Calculate the following ratios: L/L, L/L, L/L, and L/L.

b) What value does the ratio L/L approach as n approaches infinity? How does this compare to the limit of the ratio of consecutive Fibonacci numbers?

c) Prove that lim(n) L/L = , where is the Golden Ratio. (Hint: Consider the relationship between Lucas numbers and Fibonacci numbers, and use Binet's formula.)

d) For the generalized Fibonacci sequence G with arbitrary starting values G = a and G = b (where a and b are not both zero), show that the ratio of consecutive terms always approaches the Golden Ratio as n approaches infinity