Vx (1- 2 In 1 1-x/2 | + C A ! In 1 1-x/2 | + C 2. -2 In I 1-x/2 | + C -In 1 1 -x+/2 | + C O A O D E None of the Above DI

Q: )"-( 18 6. lim V n4n +(4n)" 2+ n2 4 4+ n2

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Q: In2 + 5n + 4 lim an = 1 %3D %3D 3n2 + 8n + 8 n-00 1n? In2 5n + 4

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Q: Show that (n+1)Pn+1(x) = (2n + 1)xp,(x) – npr-1(x) ;n> 1? %3D

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Q: Q5: prove 1) e2-j3n-e² = 2) |e²| = ex

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Q: 0.02x10.05x2 = -0.19 0.03x₁ +0.04x2 = 0.52 ○ X₁ = 5, X2 = 8. O x₁ = 7, 2₂ = 8 O x₁ = 8, 2 = 7 O x1 =…

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Q: Vhich among thế chôičes satist etters only. 2а. у" - Зу'- 10у %3D0 Co-5x C o2x

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Q: (a) an = }(n² + n). (b) an = 2an-1 – an-2 with ao = 0 and aj = 1. (c) an = nan-1 with ao = 1. %3D

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Q: ntiderivative. 7x x² -x-2 OB. In x + 1|-In x- 2| + c 14 .7 14 In x +1- x +1 In x-2 x- 2 3 3 14 7 OD.…

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Q: If (ar + br) 4n and n+1 k-1 E(ak - br) 1 then ar = | n+2 k-1 k-1 3 5. None 2. 72n/2

A: use elimination method Find limit for n tending to infinity.

Q: : 2 < x < 4} and B = {x € R : –4 < x < 1}. Prove

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Q: n-1 For all n > 2, 4+ 3n4" – 4n+1 - j=1 9.

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Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

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: 3. Prove the Archimedean Property, if x ER. then there exists K E N such that r SK.

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Q: (b) Define fo = 0, f₁ = 1 and fn = fn-1 + fn-2 for n ≥2. Show that 3 fak for all k ≥ 0.

A: As per the guidelines we shall solve one question only as you have asked more than one different…

Q: 6-8(/2) Ce?: x3+2°<9, x<0?\ ): ト 4,X,c03 and 年(カx2)= (4),6) = (-2x2 , 3w×2) ncludng Sfidx + dr2 = ST…

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Q: y + 5xy + y = 6 In x + 4 where y (1) = -1, y (1) = 1 -1 +x - + 20 ... 7x5 -1+ 주 - (x-1)4 + 12 (x-1)7…

A: Taylor series of a function f(x) about x=a is given as following:…

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

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Q: 10 3. Prove that the L{10e2} = s-2

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Q: let y = 3 x2 + In x - ex find y' O 6x + (1/x) + ex O 6x-(1/x) - eX O 6x +1 + ex O none of the above

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Q: (Зп+5 Find infimum of A |n E N} n+9

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Q: -In2 2e® cosh 0 do -In 4

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Q: n2 1 SHOW ALL OF YOUR WORK. 3. an = 2n +1 2n + 1" (а) Find aj, а2, аз, а4. 3 (b) Compute > a;. i=1 4…

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Q: (-1)"(n + 2) 2n+1(n + 1) 4. Compute the exact value of n=0

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Q: 9n – 5 a) lim n-- 3-n - 18 1 2n2 + 3n – 5 8n? +1 1 b) lim n- o0 || 4

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Q: 4. Find all values of x satisfying In[(2x + 1)(x +2)] = 2 In(x+2). Verify the values of x you found…

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Q: 12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show…

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Q: Show that for any n E N, n(n + 1)(2n + 1) k=0

A: The solution is given as

Q: (-1)"(n+ 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A: Given series is  ∑n=0∞-1nn+22n+1n+1

Q: cos(n") In(4) 1. 1 2. n In(n) n%3D 6n + 3 (-1)" n=1 1 4. n - 5 3.

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Q: 16. There is a very quick way to compute £-, 2'(-1)ª-' (); do use it. l=D0

A: Introduction: The binomial theorem is very important in mathematics. For any real numbers a,b with…

Q: iii) a) Prove that Pa(-x) (-1)".Pn(x) u (2n)! 22 (n!) b) Prove that P2n(0) = (-1)".- %3D

A: To prove- Pn(-x)=(-1)n.Pn(x)P2n(o)=(-1)n.(2n)!22n(n!)2

Q: () 2k = 3", Vn E N k k=0 k n+1 Σ 2 Vn E N, n > 1 3 k=2 į * () 主) n 3k = 4", Vn E N %3D k k=0 = 2"-1,…

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

Q: 15. prove that L. d= nd(n)/2

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Q: 34) Ź (K²_2K) is e (n3) Ź ķ? ビ=3 Ź 2K = k=3 ntro)(2n+1), entntt) Sungs are さ。 2n3+ 3n?+n – ant- Gn…

A: We have solve the given  problem

Q: ( Σα - xi-1) = x = x for n > 2. i=2 11 IDDLE: following "proof" th

A: Given that c, x1, x2, ..., xn, y1, y2, ..., yn are 2n+1 given numbers.

Q: n2 for all n E N. 4n Let r, :=

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Q: 3 If [E dx = In(7x - 6) - +C, find f (1). 7x - 6 Enter an integer or a fully reduced fraction such…

A: Given that:-∫f(x)xdx = ln(7x-6)-37x-6+CTo find: value of f(1)

Q: In another proof, I found the following statement. Since we are given deg f(x) = n, then the…

A: Definition: (Quotient ring) Let R be a ring and I be an ideal in R. The quotient ring is defined as…

Q: Use Gauss's trick to determine H(n)=1+5+9+13+...(4n-7)+(4n-3)+(4n+1) I got n(4n+2)/2 is this…

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Q: 5. For non-negative integer n and r, where 1<rsn, show that (n-r+1)"C,1=n(""C,). r-1

A: Given n and r non negative integers with  1≤ r≤ n. We need to prove (n-r+1)nCr-1=n(n-1Cr-1)

Q: z/(z-3)2 dz ; C: |z+3| =3 slove by residue theorem

A: The given integral is,

Q: 5. (8x*y® + 3x³) dx + (12x°y³ + 3xy²) dy = 0

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Q: 2) al X, X2 ,-.,Xn ~lid exporentíal ( 4 100/ b) Xg, Xz,=-- Xn~íid N(44i 44)=) 9= E Xi ~ ? %3=…

A: It is an important part of statistics. It is widely used.

Q: 4 and for n > 3, an 2an-1 + an-2 + 5n, express an in terms of n %3D 2. Let, a1 = 3, a2

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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: Simplification of In x + In(x + 1) = 0 is x² + x – 1 = 0; x > 0 O x² + x +1 = 0; x > 0 O x? + x-e =…

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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…

Q: Evaluate: (- +2 e + 4" – 7) dr =, +4* – 7) dr %3D | In(x) + 2e2 4* - 7x +C In(4) - - O In(x)? +e2…

A: We have to find the integral.

Q: 6n_ and If (ar + bx) = n+1 k-1 , then 1 (ak – br) = 712 n+2 k-1 k-1

A: If ∑k=1nak+bk=6nn+1 .......................(1)  and  ∑k=1nak-bk=nn+2-1n ......................(2) we…