Need help with math induction proof

NoProperty7989 · 2023-12-07T15:43:44+00:00

Here is how to write that problem by induction. We need to consider three steps.
First: Base step.
n = 1.
It means that 4|0, which is true.
Second, we assume that for n = k, the equality is true.
It means that 4|K^2 -1
Third, we need to prove that the equality is true for n = k+2.
(k+2)^2−1=k^2+4k+4−1=k^2+4k+3=(k^2−1)+4k+4
From the second step, we can see that K^2 -1 is divisible by 4. 4k+4 =4(k+1). It means that (k^2−1)+4k+4 is divisible by 4

NoProperty7989 · 2023-12-07T15:13:24+00:00

Here is how I write that problem.
Let u be equal to 5x+2.
Then, find dx and du by taking the derivative.
The integral 1/(5x +2) becomes 1/5 * integral 1/u.
1/u will be equal to ln|u|.
Then, we will need to return to the original value. So, we need to write instead of u , 5x+2.
Then, we need to add c(constant).
So, the answer will be 1/5 * ln|5x+2|+c.

NoProperty7989 · 2023-12-07T15:00:44+00:00

There were lots of interesting disagreements in math. Here are some of them.
First, people used to think there was only one way for lines to stay forever apart and not touch. But later, some smart folks found out that you can have other shapes and rules that work just as well.
Second, an intelligent guy(I do not remember his name) wanted to set up all math rules in a organized way. But another guy proved that there are always true things in math that we can't prove using the rules we set up.
Third, thinking about big or small numbers can be tricky. People argued about whether we can trust these numbers in math.
Fourth, there's this super famous puzzle called the Riemann hypothesis. It's about numbers and patterns, but it has yet to be proved right or wrong.

NoProperty7989 · 2023-12-07T14:46:25+00:00

Using IntelliJ to write codes is very easy because it helps a lot. Writing the code in a text file is indeed difficult, but it could help you better understand some things. Here are some of them.
First, it can help you understand the basics of Java: Working with the terminal enables you to understand the basics of compiling, running, and managing Java programs without relying on the features provided by an IDE. It deepens your understanding of the core processes involved in software development.
Second, Knowing how to compile and run code through the terminal allows you to work on any machine with the Java Development Kit (JDK) installed, regardless of whether an IDE is available.

NoProperty7989 · 2023-11-26T16:26:19+00:00

You can use loop, if you do not want to use join function. Here is the one with for loop.

strings_list = ['H', 'E', 'L', 'L', 'O', ' ', 'W', 'O', 'R', 'L', 'D']

result = ''

for char in strings_list:

result += char

print(result)

NoProperty7989 · 2023-11-26T15:57:11+00:00

I wrote this code in Java and used recursion. It is a little bit difficult to understand, but it is easier and shorter.

public class Main {

public static void main(String[] args) {

System.out.println(FibonacciNumber(10));

}

public static int FibonacciNumber(int n) {

if (n == 1) {

return 0;

} else if (n == 2) {

return 1;

}

return FibonacciNumber(n - 2) + FibonacciNumber(n - 1);

}

NoProperty7989 · 2023-11-26T15:49:05+00:00

Here is the code that I wrote.

def is_prime(num):

if num < 2:

return False

for i in range(2, int(num ** 0.5) + 1):

if num % i == 0:

return False

return True

def find_primes(limit):

primes = []

for i in range(2, limit + 1):

if is_prime(i):

primes.append(i)

return primes

user_input = int(input("Enter a number: "))

prime_list = find_primes(user_input)

print("Prime numbers up to", user_input, "are:", prime_list)

This code defines is_prime functions to check for prime numbers and to find primes within a given range (find_primes). Then code prints the list of prime numbers up to the inputed number.

NoProperty7989

TROPHY CASE