46. What is the coefficient of p³q!2 in the expansion of (p + q)15? А. 345 В. 455 С. 565 D. 625 47.In the expansion of (2x +2) there is a term that has no x in it. If that term is equal to 2,689,120, what is the value of k? А. 5 В. 6 C. 7 D. 8

Q: What is the coefficient of x²wa³b in the expansion of (x+y+w+a+b)??

A: To determine the coefficient of x2wa3b in the expansion of x+y+w+a+b7.

Q: 20 Term independent of x in expansion of x2 5 + 6x3 is? e 7

A: Here we have to find the term independent of x.

Q: 16. The expansion of (2 + 3x;-1 is valid when >x>

A:

Q: value of ∑n=1∞(6an+9bn) if ∑n=1∞an=8 and ∑n=1∞bn=2.

A:

Q: D) Sebastian has worn a black shirt on 8 of the last 20 days. What is the experimental robability…

A: P(A) = n(A)n(T)

Q: What is the coefficient of b° in the expansion of (6 -2b)17 O a. -b * C(17,9)* 29* 68 O b. C(17,8)*…

A: binomial series expansion

Q: 6. A. Show that 81 – x²dx : 81n %3| 4

A: According to company rule we can answer only first question and rest can be reposted

Q: ity by 11 by alternately adding and subtracti left, beginning with the units digit-that is, units…

A:

Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

A:

Q: Show that n!> n3 if n is large enough.

A:

Q: 1 Prove 2.5 1 1 + .…+ 8.11 1 for n > 1. %3D 5.8 (3n-1)(3n+2)

A: Prove S(n) for n=1.   LHS =1/2.5 =1/10; RHS = 1/6+4=1/10   So it is true for n=1.

Q: . Show that n! > 3ª for n> 7.

A:

Q: 3. What is the value of the 6th term of the expansion (a + b)? O 84a 66 O 126a 64 O 126a*b5 84a 3

A: Given: (a + b)9.

Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: Show that if 5 divides n2 with remainder 4, 5 divides 3n3 with remainder 2.

A: Given that 5 divides n2 5 n25 n×n5 n5 n3 As it gives the remainder 4 on dividing by 5. Therefore it…

Q: Show by mathematical l Induction y 4n-2 3. +17 can be divided by +22 possitive integers 16 far all

A: To prove the given statement by using mathematical induction.

Q: What is first 5th term of a1= 5 and r=3

A:

Q: 24 + * ? 8 What is the coefficient for the yl8 term in the expansion of (5x°y²

A: We use the Binomial Theorem to answer the given question.

Q: What is the coefficient of x^8 in the expansion of (2+x)^11?

A:

Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: 2. What is the value of the 3rd term of the expansion (3y- 2)'? | O - 10206y O 35a+b3 20412y5 O 21a…

A:

Q: 1) What are the values of C and k of the term Cykx20 in the polynomial expansion of (x² + y )100? 2)…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 21 In the expansion of (x +y), write the value of the exponent k in the term that containsr2k--4,,3k

A: In the expansion x+y21, we have to write the value of the exponent k in the term that contains…

Q: 8. [8] What is the coefficient of y* in the expansion of (3y-2)'.

A:

Q: 6. Find the coefficient of x* in the binomial expansion of| 3 , if exists. 4 2.x

A:

Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

A:

Q: If in the expansion of (1 + n}+", the 5th term is 24 times the 3rd term, fnd the value of n.

A:

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

Q: Show that (n + 1)! > 2"+3, for n > 5.

A:

Q: 29 Using the combination formula expand (5+x)

A:

Q: Suppose that 2% of a population carries antibodies to a virus. For a research project, you need 8…

A: Given that 2% of the population carries antibodies to a virus. Hence the probability that a person…

Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

A:

Q: 2. Prove by the Principle of Mathematical Induction that 5-2 + 5-4 + 5-6 +... + 5-2n = 24-'(1 –…

A:

Q: 30. Find the term containing y12 in the expansion of (b + y*)". A. 25b³y12 В. 42b8y12 C. 35b*y12 D.…

A:

Q: Show that 2^n >n^2 if n is large enough.

A:

Q: Show that n! > 3ª for n > 7.

A: The given inequality is, n!>3n We need to prove that, the inequality is true for n≥7 Firstly,…

Q: 1. Show that b, = (n!)'/n > 20.

A: Consider n!1n=an1n.

Q: 2n Using the expansion of (1 + x)² and given that 2nC₁ + 2(²nC₂) + 3(2n C3) + ... + 2n(2nC₂n) +…

A:

Q: 15 Use the Binomial Theorem to expand: 2x 80x4y , 80x³y?, 40 10 A. 32x+ 3 9 27 + 81 243 80x4y, 160…

A:

Q: 27. What is a simplified expression for the 4th term in the expansion of (3x2 +4y)® written in…

A: The given expansion is (3x2+4y)9

Q: 3) Find the Term that includes x? of the expansion (x +

A:

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: -)) In how many days does 7 masters paint the wall painted by 5 masters in 14 days

A:

Q: 5. What is the coefficient of x''y" in the expansion (- 3x + 4y)ª . Show your work here:

A:

Q: 8. Show that no prime number of the form 4k +3 can divide a number of the form n2 + 1.

A:

Q: 04 Prove that m (n-k) = (n+1)- ("-m)n 2m n2 m 2

A: Note: Cr+1n+1=Crn+Crn-1+…+Crn-m+Crn-m-1+…+Crr→1 From identity 1, Crn-m=Crn-m-1+Crn-m-2+…+Crr Now,…

Q: for n greater or equal to 1  proove that 2+4+6+...+2n=n^2+n by using mathematical induction

A: