2n-1 (2j + 1) = 3n² for all positive integers n. Prove that j=n

Q: 1. Prove that 1 {2" : n E Z} = {; :n e Z}. 2n

A: I used the fact that two sets are equal iff they are subsets of each other

Q: solve the recurrence relation with the given initial condition.

A: y1 = 1 , y2 = 6 , yn = 4yn-1 - 4yn-2 for n ≥ 3 y3 = 4y2 - 4y1 = 24 - 4 = 20 y4 = 4y3 - 4y2 = 80 - 24…

Q: Prove by mathematical induction that, for all positive integers n, de (e* sin x) = 23"e* sin(x…

A: To Prove →dndxn ex sin x = 212n ex sinx + 14nλBy mathematical induction for all positive integen…

Q: x2+1 =0 Has soluion in R

A: Solution:

Q: Let n E Z. Prove that if 3/ (2n² + 1), then 3 | r n.

A: Solution:     Let 3 does not divide n      => n = 3k ± 1 for some integer k.       => 2n2 + 1…

Q: дгу ду дх2 5- · + 4y = 0 дх In

A: Given differential equation is y''−5y'+4y=0.....1. We have to find the solution of the expression…

Q: s expansion of 2 , valid in the re of 1/z2 is:

A:

Q: (a) Show that: (i) dim R" = n (ii) dim R2×2 (iii) dim Pn = n+1 = 4

A: Basis of a vector space: Let us consider a set S⊂V(F) is said to be basis of vector space V over…

Q: Evaluate the integral 12 +1 -dt 3 (213 + 6t + 1) The integral is -1/(12*2^3+6x+1)^2+((1/12))

A:

Q: diverges because an > 0. for all n 2 1. 2n + 1 True False

A: Given: The statement "an=n2n+1diverges because an>0 for all n≥1." To determine: Whether the given…

Q: Prove that A=I-N+N²-N³ for A & N definec as follows: N=A-I (identity), where |1 0 0 an & I= 0 1 0. 0…

A: given matrix  A=1a12a1301a23001 and I=100010001 prove  that A-1=I-N+N2-N3 where N=A-I

Q: 1. Using (2n +1)Pn= Pn+1-P-1: prove that P(x) = (2n-1) Pn-1(x) + (2n-5) Pn-2+(2n-9) Pn-3+.... - and…

A: Given : 2n+1Pn=Pn+1'-Pn-1' To Prove : (a) Pn'x=2n-1Pn-1x+2n-5Pn-2+2n-9Pn-3+ . . . . . .…

Q: The solution of e² = -2 for n = 0,±1,±2,. is %3D ... Select one: O a. In2 + in(1+n) O b. In2 +…

A:

Q: 5) Show that n(n+1)(4n-1) E"=1 r(2r – 1) 6 Hence find E015 r(2r – 1) |

A:

Q: Show that F0F1 . . . Fn−1 = Fn − 2 for all n ≥ 1, where Fi is the i-th fermat number

A: To Show -

Q: Q4: 1- Prove that (1-0+(1+ () 0. 2. Put in the form a+ib ((1+i))- ((1-i)).

A: Here we have to prove that, 1) 1-i2+(1+i)2=0 2) Put in the form of a+ib, 1+i22-1-i22

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

Q: Prove that L ={(1Z*)"10" | n e N}is not regular.

A: The given language: Then, the above language can be rewritten as, To prove that L is not regular.…

Q: Eii!): i(i!) = (n+1)! – 1, for every integer n > 1. i=1

A:

Q: (b) Prove that for any positive integer n, j. 2³ = (n – 1)2"+1 +2 j=1 (c) n(n? – 1) - Prove that for…

A:

Q: 4. Use mathematical induction to prove that V n E Z* U {0}, [I²-o(2i + 1)! 2 ((n + 1)!)**. n+1

A:

Q: :Find the first 3 terms for (1 (-1)*1 (2n – 1)!J n+1

A: We have to find the first three term of the given series.

Q: 28. Prove by induction that i(i – 1) (n+1)n(n– 1) i=1 holds Vn E N,n 2 1.

A:

Q: Theorem 13.Let {Xn} be a solution of Eq.(1). Then the following statements are true: (1i) b 0, the…

A: To get the inequality (49). XN+1≥bdA+B+C+D+b2d2−be               .......................49 We have…

Q: 68. Let Jr dx. %3D (a) Show that J1 = -e-x²/2. (b) Prove that Jn = -x"-le-x*/2 + (n – 1)Jn–2. %3D…

A: Solution is given below:

Q: :find E10 (-1)(n-1) Zn=1 (n+1)2 %3D1

A:

Q: 1. Condense and expand. a) Condense as a single quantity: 1 In(3)+- In(x? +1)-3 In(1+x) 3 b) Expand:…

A: Given, (a)  To condense ln3 + 13ln(x2+1) -3ln(1+x) (b) To expand lnx2+1x+2(1+x)5

Q: 5. (a) Find the real constants P, Q, R, for which 3 P Qx + R x2 – x + 1° x3 + 1 x + 1 | Show that V3…

A: 5(a)  3 / (x3+1) = P/(x+1) + (Qx+R)/(x2-x+1) ......(1) Now, 3/ (x+1)(x2-x+1) = P/(x+1) +…

Q: Find the sum of f(i) = 2i +2 over the integers 1, 2, 3, ..., 20.

A: Given,           The sum of fi=2i+2 over the integers 1,2,3,...,20

Q: Find the nverse 2 -1 2 A =|1 0 1 f it enists. 0 3 -1

A: The given matrix is A=2-1210103-1. To find the inverse matrix, augment it with the identity matrix…

Q: Theorem 13.Let {xn} be a solution of Eq.(1). Then the following statements are true: (i) b 0, the…

A: To get the inequality (48). XN+1≥bdA+B+C+D+bXN−kdXN−k−eXN−1               .......................48…

Q: Consider z1 = 1- j2 and z2 = -2 + j3. What is the result of 5z1 + 3z2? %3D -1 -j O 1-j none of the…

A:

Q: Let x1 V3, xn+1 = V3*" ,n > 1. Use induction to show that x, < 3 for all n E N.

A: Given x1=33,  xn+1=3xn3. Show that xn≤3 by using induction for all n∈ℕ.

Q: Solve a0 = 1, and an = 3an−1, for n ≥ 1 using the characteristic equation method.

A: We will start with the trial solution, an = rn to find the characteristic equation.

Q: Differentiate f(N) = (N) = 0 bN² + N with respect to N. Assume that b and K are positive constants.…

A:

Q: 2. Prove 1+5+9+13+ .…+ (4n – 3) = n(2n – 1) using PMI. ... %3D

A:

Q: 4. Evaluate det 210 TONO -4 0 2 5406 -2 by cofactor expansion about a row/column of your choice.

A:

Q: Verify that the Diophantine equation x² + y² where d'(n) and d"(n) are the numbers of positive…

A: We will first show that product of two square numbers can be expressed as a sum of two square…

Q: (iii) Prove that there is a constant C such that 0(2º) 1.

A:

Q: { (3i – 2)² = -n(6n² – 3n (b) Prove by induction that | i=1

A: We have to prove that ∑i=1n3i−22=12n6n2−3n−1 for n≥1 by using principal of mathematic induction.…

Q: Find the residue of 3 Z 2 z² - 1 at z→ ∞0.

A: We have to find the solution of the given problem.

Q: The fourth term in the expansion (2r + y)' is

A: The equation for the combinations n taken r at a time is equal to Crn=n!n-r!r!.  The continued…

Q: Find o v' (+2) dt. Show the computations that lead to your answer

A:

Q: 3. Prove that A=I-N+N²_N³ for A & N defined as follows: N=A–I (identity), where 1 0 0 & I=|0 1 0 1…

A:

Q: Q2\ Prove that Ax (B+C) = A×B + AxC When A= 2i + j-k B=i-j+ 2k Sumer C=i+2j+k

A:

Q: 2) × 2+4i 3+1 and 2=3-1 2+i 2 prove that x² + 3xy + y² = 6 if it was

A:

Q: 1. For positive integers i, j,n such that i+ j = n (a) show that corresponding multinomial…

A: sol(a) we know that multinomial co-efficient is definedby ni,j=n!i!,j!  —(1) where i+j=nnow…

Q: 5. Prove that |e³ – Eosisn l < 32 3n+1 e3 (n+1)!

A: The given problem is to prove the following summation inequality. We have to use taylor series for…

Q: 6. Inductively prove that B(0) = 0 for all n. Conclude that B(2) is C° function. %3D a

A: We prove the above theorem by induction. We have to prove that Bn0=0 ,∀n...1 Take n=0 then, Bn0=B0=0…

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.