Let un be the nth Fibonacci number. Prove that the Euclidean algorithm takes precisely n steps to prove that gcd(un+1, un) = 1.

Hello, Please consider the following.

Fibonacci numbers are defined as

Step 1 - The proposition is true for n = 1

Let's apply Euclidean algorithm

This is exactly 1 step and

Step 2 - We assume that this is true for k, meaning that the Euclidean algorithm takes precisely k steps to prove that

So, the first step of the algorithm is

And from now, we need to find which is 1 and takes k steps, by induction Hypothesis. Therefore, it takes k + 1 steps to find

Step 3 - Conclusion

We have just proved that the Euclidean algorithm takes precisely n steps to prove that

Thanks

38.5

step-by-step explanation:

lets say, first number = r

second number = 3r+7

total = 49

(r)+(3r+7)= 49

4r+7 = 49

4r = 42

r = 42/4

r= 10.5

3r+7 = 3(10.5)+7 = 38.5

hope this

solution:

→c= a b , that is product of a and b is possible only when when a is m × n matrix and b is n × p matrix.

that is, number of columns of a= number of rows of b.

so, size of matrix , c= a b is , if a = (m × n)matrix

b= (n × p)matrix

then a b = [ (m × n)matrix] ×[   (n × p)matrix ]

→→c= a b= [m × p ] matrix

→→size of product a b will be equal to = number of rows of a × number of columns of b