(d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

Q: 3n > n for every positive integer n.

A: Now, since we assumed 3k>k, we can substitute k with 3k in the inequality: 3k+1>3k Since ,…

Q: P(X₁ = x₁, ..., Xn = xn) = P(X₁ = x₁) [I²²2 P(X; = Xi| X₁ = X₁, ..., Xi−1 = Xi−1 )

A: Here we have a given equation that

Q: calculate 1+3+5+...+(2n-1) for n = 1,2...,6.

A:

Q: determine the value of c(n,5) /p(n,5) where n can be any integer thats 5 or greater

A: Recall the following. Permutations Number of permutations of n things taken r at a time: pn,r=n!n-r!…

Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: Expand and simplify    (i) (2k − 3)(k − 2)

A:

Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

A:

Q: a. 3.2F> Fn+1 for n23 [Solutions and final ans

A: We need to prove 2Fn>Fn+1 Now, we have formula for fibonacci sequence is Fn=Fn-1+Fn-2

Q: Find the value of  n! (n − r)!r!  when n = 6 and r = 4.

A:

Q: For n = 10 and x = 7, find n! x!(n-x)!

A: The given expression is .To solve the expression for and .

Q: 00 8tan-in Zn=1 n3D1 1+n2

A:

Q: Prove that 2n + 3n is a multiple of 5 for all odd n € N.

A:

Q: . Verify the formula п(n + 1) 2 for n = 10, 50, and 100.

A: ∑x=1nx = n(n+1)2To verify the formula for n = 10, 50 and 100.

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

A:

Q: Which expression is equal to E (2" + n²) k-1 n=2 O (2" + n?) +D ¬k-2 (2" + n2) n3D2 nk-2 O (2*-1 +…

A:

Q: For every integer n > 1, 2+4+6+ + 2n = n² + n.

A:

Q: Let A L(R). Suppose all solutions of x' = Ax are periodic with the same period. Then A is semisimple…

A:

Q: Which of the following is true? Select one: a. (n+1)= n(n) for any real number b. г(n) = n(n+1) for…

A: The objective of the question is to determine which of the given mathematical statements is true.

Q: (d) an 10"+1 +3 x 4"-1+5 is divisible by 9 for all n2 1,

A:

Q: For which values of n does n! have n or fewer digits

A: Solution is as follows

Q: 4. Find the first three terms of (3x – 2)20. What is the coefficient of æ10?

A:

Q: #5 Find An if n is a positive integer and A = 4 -3 - 2

A:

Q: Show that if n is a positive integer, then fo(n) (26(n) $(2n) = if n is odd if n is even.

A: Given- n is a positive integer. To prove- ϕ2n=ϕn     if n is odd2ϕn   if n is even   Result Used-…

Q: If n is an odd integer, then the floor of n/2 = (n-1)/2.

A:

Q: For each integer n>1, let A(n) denote the sum of the integers from 1 to n. For example,…

A:

Q: What is the smallest integer n such that d(n) = 8? and d(n) = 10?

A: