Show that for every integer n, f (n) = (2 n3 + 3 n – 10 n + 5 n) is divisible by 7. %3D (Hint: use the Fermat's Theorem).

Q: Find the smallest integer x for which (x4 + 4x)/(x + 2) is O(xx). 01 02

A:

Q: (1+ 2x + 4x + 8x° + · · · )(3x + 3x* + 3x° + · · · )(x° + 2x° + 3x° + •..)

A:

Q: (1+ 2x + 4x? +8x + ...)(3x² + 3x* + 3a° + · . .)(x³ + 2a® + 3x° +

A:

Q: For all n ≥ 1, let an = (a) Find a1, a2, A3, A4. (b) Computea. i=1 (c) Computea. i=1 (n + 1)² n!…

A:

Q: 1. Use the binomial theorem to expand (a) (1+x)* and (b) (1+x). 2. Use the binomial theorem to…

A:

Q: 10. Use the binomial theorem to evaluate (2r³ - y) 5.

A:

Q: The coefficient of x²y4z5 in the expansion of (3x+y+z)¹¹ is

A: The given expression is: (3x + y + z)11 To find: The coefficient of x2y4z5 in the expansion of (3x +…

Q: Show the function f1(x) is primitive recursive. f1(x) returns n if factorial(n) = x but returns 0 if…

A:

Q: Find the 5th term in the expansion of x3 7 2y2) x2

A: The expression is  x3-2y2x27

Q: Jack has a large baseball card collection of 2640 cards. He has them organized alphabetically from…

A: We have to find

Q: i. Find the coefficient of x in the expansion of ( 2x –÷) . ii. Hence find the coefficient of x in…

A:

Q: Show that when |z +1| < 1,z² =1+£(n +1)(z+1)" n=1

A:

Q: What is the rule for this pattern? (figure 1 is 1 block) Of(x)=n+2 Of(x) = 2n Of(x)=n-1 Of(x) = 2n-1

A:

Q: Prove that product notation d/n = nτ(n)/2.

A:

Q: (c) Find the term independent of x in the expansion of, 15 + 4x?

A:

Q: find the Maclaurin expansion of e-2x In(1+x)

A: fx=e-2xln1+x

Q: 10 [1] Find the coefficient of the term involving x* in the expansion of (3x - a. -262444 b. -262440…

A: We have to find the coefficient of x^4.

Q: 12. Use the Binomial Theorem to expand the expression (2x-3y). Show your work.

A:

Q: Prove the following using the binomial theorem: ()=()+ 2n

A: In this context, the aim is to prove 2n2=2n2+n2.

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: Use the binomial theorem to expand the expression (u - 5v)¹

A:

Q: 2021 2. Expand (1 + v3i)“ recall the binomial theorem.

A:

Q: Find the smallest positive integer N≥ 2 that gives remainder 1 upon division by each of the integers…

A:

Q: Use induction to prove that (2n + 1)² – 1 is divisible by 8 for each positive integer n.

A:

Q: Given (xi , xi2022 i ) ∀i = 0, ..., n, where x0 = −1, x1 = 0, x2 = 1, x3 = 2, ..., xn = n − 1. Write…

A: Given xi,xi2022∀i=0,1,2,...,n  Where x0=-1, x1=0, x2=1, x3=2,..., xn=n-1

Q: Write the recursive definition for an = pn n! where a1 = p.

A: The recursive formula is used to describe the relationship between two terms of any sequence. The…