Question 4 For any integer k > 0 and any integer j, we can define () j·(j–1).…(j-k+1) (Notice what happens when j < k!!) You can assume that k! Pascal's identity holds for this extended version of the binomial coefficients. Proue that Σ, () n+1' (k+1. j=0

Q: 1. In your own words, what is the binomial theorem. Find the 5th term of (x+7)

A: Given : x+77

Q: I(f) = μετα fi + 3 +3 N-1 Σ fitfiti + + 2 i=2,5,8,... N-2 Σ i=4,7,10,... f; ] + fN+1

A: Given: Objective: To write the above formula in MATLAB.

Q: 13. Floor and ceiling notation were defined in Section 1.1. (a) Show that []+[=n for all positive…

A: Floor and ceiling.

Q: 7. Suppose that {an) is defined recursively by an = a-1-1 and that ao = 2. Find a3 and 44.

A: Solution :-

Q: What are the coefficients of æ5000 and æ4999 in the simplified expansion of (5æª – 2)²2021?

A:

Q: Question 11 (Зг+1) da x2-a-6 Evaluate: O None of these O 2 In a-3+In | a+2 |+C O In a-3+2 ln | a+ 2…

A:

Q: If P(A)=0.2, P(B)=0.3 an d :P(ANB)=0.1, then P(A'UB) is

A: As per bartlby guidelines we can answer only first question and rest can be reposted  Given:…

Q: Prove by induction that, for every n\geq2 n+1 (1-2)(1-¹)(1-(1-1/2) = ² 2n

A: To Prove: We have to prove by induction that for every n≥2, we have:…

Q: Зх —1 A B (b) (i) Express (1– x)(2 - 3x) in the form - x 2-3x , where A and B are integers. Зх— 1…

A:

Q: 6 Without expanding, simplify: a 1 + 3(x – 1) + 3(x – 1)² + (x – 1)³ b 1- 6(x + 1) + 15(x + 1)² –…

A: Recall: (a+b)3 =a3+b3+3a2b+3ab2 (a+b)n=C0n an b0 + C1n an-1b1+C2n an-2b2+...+Cnn a0bn

Q: 2. What is the general term of 1, 3, 6, 10, 15? O 2n – 1 n^2 – 1 (n^2+1)/n O (n(n+1))/2

A:

Q: for

A:

Q: 11. Home Work. Use Binomial Theorem to find the coefficient of 232 in the expansion of (a¹ + 3)¹5 1…

A:

Q: (b) Prove that for every n e Z, n³ +n is even. (c) Prove that for all ne Z, n is odd if and only if…

A:

Q: Q. 4. Convert the number(4215.245), to octal. Q. 5. Show that "-'C. r-1 11 2 Q. 6. Find the…

A:

Q: Use four terms of Newton's binomial expansion to approximate 11.

A:

Q: Question 4 For any integer k > 0 and any integer j, we can define () j-(j–1).(j-k+1) k! (Notice what…

A: We know that, jk=0 when j&lt;k

Q: Question 7 Expand the expression (Vī -) using the Binomial Theorem. Simplify all coefficients and…

A: In the given question we have to expand the expression                                     by using…

Q: The multiplication rule says that if an operation can be performed in k steps and, for each i with 1…

A: Answer is mentioned below

Q: 3. From Eq. (1), deduce that U2n-1 2. U2n 2. %D %3D n 2

A: 3

Q: Question 3 Prove or disprove: 31 (2n² + 1) if and only if 3 m 3| n

A: Any integer can be written in form of 3k or 3k+1 or 3k+2 for some integer k

Q: Question 2 By drawing Pascal's Triangle and applying Pascal's Identity, show that both methods may…

A: We can solve this by Pascal triangle formula as well as Pascal identity

Q: 5. Assume n > 3 is an integer. n!(n +1)! (n – 1)!(n +2)! (a) Simplify

A:

Q: 50 Evaluate Σ (5 + 2i) i=1

A: Solution is done in step 2

Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: 16. Find the term containing x18 in the binomial expansion of (x2- y)15.

A: A binomial expression is an expression of the form a+bn. This binomial expression can be expanded…

Q: a) Suppose that in the binomial expansion of (1 + m2)", where m and Kare non-zero constants, the…

A:

Q: 12. Suppose that {an} is defined recursively by a, = a - 1 that ao = 2. Find az and a4.

A: Given relation is an = an-12 - 1 With a0 = 2

Q: 12 - [18 = 3] 10 Find formulas for the entries of M", where n is a positive integer. M = . Hint: a…

A:

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

A: Solution is as follows

Q: 10. a) Recursively define ao = 1, a; = 3, a2= 5 and an = 3an2 + 20ars for n2 3. Çalculate.gafor n =…

A: