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Recurrence and ...

Requrece

Task 1

Compute first 5 elements of the following sequences:

  1. \(a_0 = 1\), \(a_{n+1} = 2a_n + 1\) for \(n \in \mathbb{N}\setminus\{0\}\).
  2. \(b_0 = 2\), \(b_{n+1} = b_n^2 - 1\) for \(n \in \mathbb{N}\setminus\{0\}\).
  3. \(c_0 = 2,\ c_1 = 3\), \(c_{n+2} = c_{n+1} \cdot c_n\) for \(n \in \mathbb{N}\setminus\{0,1\}\).
  4. \(d_0 = 1,\ d_1 = 2\), \(d_{n+2} = d_{n+1}/d_n\) for \(n \in \mathbb{N}\setminus\{0,1\}\).
  5. \(e_0 = 1,\ e_1 = 2\), \(e_{n+2} = e_{n+1} - e_n\) for \(n \in \mathbb{N}\setminus\{0,1\}\).

Task 2

Define folowing formulas and sequences using recurence:

  1. \(n!=1\cdot 2\cdot 3\cdot \ldots \cdot n\) for \(n\geq 1\).
  2. Fibonacci numbers.
  3. Napier's number
  4. \((2,2^2, (2^2)^2,((2^2)^2)^2,\ldots)\)
  5. \((2,2^2, 2^{2^{2}}, 2^{2^{2^{2}}},\ldots)\)

Sequences and series

Task 1

Calculate:

  1. \(\frac{7!}{5!}\)
  2. \(\frac{10!}{6!4!}\)
  3. \(\frac{9!}{0!}\)
  4. \(\sum_{k=0}^{5} k!\)
  5. \(\prod_{j=3}^{5} j\)

Task 2

Simplify the fractions:

  1. \(\frac{n!}{(n - 1)!}\)
  2. \(\frac{(n!)^2}{(n + 1)!(n - 1)!}\)

Task 3

Calculate:

  1. \(\sum_{k=1}^{n} 3^k\) for \(n = 1, 2, 3, 4\)
  2. \(\sum_{k=n}^{10} k^3\) for \(n = 3, 4, 5\)
  3. \(\sum_{j=1}^{n} j\) for \(n = 1, 2, 5\)

Task 4

Calculate:

  1. \(\sum_{i=0}^{\infty} (-1)^i\)
  2. \(\prod_{n=1}^{\infty} (2n + 1)\)
  3. \(\sum_{k=3}^{8} (k^2 + 1)\)
  4. \(\left(\sum_{k=3}^{8} k^2\right) + 1\)

Task 5

Calculate:

  1. \(\prod_{n=1}^{m} (n - 3)\) for \(m = 1, 2, 3, 4, 73\)
  2. \(\prod_{m=1}^{k} \frac{k+1}{k}\) for \(m = 1, 2, 3\). Provide a general formula for the product for all \(k \in \mathbb{P}\).

Task 6

Calculate and find the general formula for:

  1. \(\sum_{k=0}^{2} k^2\) for \(n = 1, 2, 3, 4, 5\)

Task 7

Consider the sequence defined by \(a_n = \frac{n - 1}{n + 1}\) for \(n \in \mathbb{N}_{+}\).

  1. Write the first six terms of this sequence.
  2. Calculate \(a_{n+1} - a_n\) for \(n = 1, 2, 3\).
  3. Prove that \(a_{n+1} + a_n = \frac{n(n+1)}{(n+1)(n+2)}\) for \(n \in \mathbb{N}_{+}\).

Task 8

Consider the sequence defined by \(b_n = \frac{1}{2} \left(1 + (-1)^n\right)\) for \(n \in \mathbb{N}\).

  1. Write the first seven terms of this sequence.
  2. What is the set of all values of this sequence?

Combinatorics

Variations without repetition

  1. From five different balls, choose two and arrange them in a specific order. How many arrangements are possible?
  2. You have four different books. Choose two and arrange them on a shelf in a specific order. How many such arrangements are there?
  3. In a competition with 10 participants, how many ways can 3 finalists be selected and arranged in order?
  4. You have seven different keys. Choose three and arrange them in a specific order on a table. How many possible arrangements are there?
  5. At school, three people are chosen as leaders, and their roles (chairperson, deputy, and treasurer) are significant. How many different arrangements can be made for 8 candidates?

Variations with repetition

  1. You have three different colors of paint. How many different sequences of two colors can you create if colors can repeat?
  2. The menu includes four different types of drinks. How many different two-drink combinations can you create, allowing repetition?
  3. A lock code consists of three digits, each ranging from 1 to 5. How many different codes can be created if digits can repeat?
  4. In a class, you have 6 different pens. How many different combinations of two pens in a specific order can be made, allowing repetition of the same pen?
  5. A shop offers three types of candies. How many different sets of two candies can be created if you are allowed to pick the same candy multiple times?

Permutations without repetition

  1. How many different three-digit numbers can be formed using the digits 1, 2, 3, 4 if no digit repeats?
  2. You have 5 different letters: A, B, C, D, E. How many different words can you form using all of them?
  3. In a group of 6 people, how many different ways can they be arranged in a line?
  4. You have four different pictures. In how many different ways can they be hung on a wall in a row?
  5. At school, there is a contest where 7 students need to line up in a specific order. How many different arrangements are possible?

Permutations with repetition

  1. How many different words can be formed using the letters in the word "ANNA"?
  2. You have a box with three red balls and two green balls. How many different orders can these balls be arranged in?
  3. In the word "COCONUT," there are three "O"s. How many distinct words can be formed by permuting the letters of this word?
  4. In a group, there are 3 people named "Adam" and 2 people named "Eve." How many different arrangements of this group are possible?
  5. How many different numbers can be formed using the digits: 1, 1, 2, 2, 2?

Combinations without repetition

  1. You have 10 books and want to choose 4, but the order of selection doesn’t matter. How many possible selections are there?
  2. In a class of 15 students, how many ways can you select three for a trip?
  3. From a standard 52-card deck, you choose 5 cards. How many different hands can be formed?
  4. From 8 different fruits, choose 3 without considering the order. How many such selections are there?
  5. A company has 20 employees. How many ways can a 5-person team be chosen?

Combinations with repetition

  1. A shop offers 4 different types of fruit. How many different sets of 3 fruits can you buy, if the same fruit can be selected multiple times?
  2. You have 5 different types of candies. How many different sets of 4 candies can you choose if candies can repeat?
  3. In a restaurant, there are 6 different desserts to choose from. How many different two-dessert sets can you create if you can choose the same dessert twice?
  4. A flower shop offers 3 different types of flowers. How many bouquets of 5 flowers can be made if flowers can repeat?
  5. You have 7 different ice cream flavors. How many different 4-flavor combinations can you create if flavors can repeat?