6. Prove the following identity: n ()-(7)-(-)) (" + ¹) + (x − ¹). k k-1, k

Q: Determine L-1) 9s – 8 (? – 2sJ*

A:

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

A:

Q: Which of the following is false? O ("2) = 2(") k ㅇ ") %3 (4") + (F) k n(n+1) (":"): 2 2 O ()² = (?)…

A: We have to check whether given statement is true or false.

Q: Let {(=) = In ="(z +2)*(z² + 9)] f'(x) =

A:

Q: 2 If C = |2 -1 0 Show that (C")-1 = (C-')" -2 2 2

A:

Q: 5. (a) If r23 = 0, prove that R₁ (23) = (23) = 712 + 7³1 01-23 = (1 - 12 - 13). 07.

A:

Q: y² + y²e6)° x' 2 + 2x²eG)* , 2xye 2xye 6)*

A:

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: 8. The (2*+1 – 2k) = %3D k=0

A:

Q: 133. -Σ 2" Sn+ -(x + 2)" n=0

A: Consider the series, Rewrite this as Now take the constant outside,

Q: Prove that _f₁ = = ₁+2 +851+1 −85₁-1 + ₁-2 +0(hª) 12h

A: Use taylor series expansion to prove this

Q: Show that (p-1)(p2) (pr) (-1)'r!(mod p), for r= 1, 2, . . . ,p - 1. .

A:

Q: 5. Prove that, for n> 1, n

A:

Q: 5.3.2 (a) Suppose that n > 3. Prove that ("+1)ʻ 5.

A: Given:-             n+1n2 < 2 n+1n2 -22<0n+1n+2n+1n-2<0n+1n+2 =…

Q: -Prove that a) b) n -("71) = (-+ 0 (," }) n r r+ n 2-0 (7) = 0 r=0

A: Also we know that    Binomial theorem

Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

A: .

Q: 81. Prove: 2+4 +6+ + 2n = n(n + 1). ...

A:

Q: 1. +1 Select one: a. th n! b. tn-1 (n 1)! C. n! tn+1 d. n! th-1 e. t+1 (n +1)!

A: Solution:-    L-11sn+1  ∵we know  that:-By using Laplace properties:-

Q: en-3h-S If y= show that day (on-3)* 2n-3

A: Given function is  y=ex^2-3x-52x-3                           ⋯1 To show:…

Q: 5. Prove the following identity for n 2 1: Fzn+1 = F+1+ F;

A: We need to prove that ,  F2n + 1 = F2n+1 + F2n We know that ,   Fm+n = FmFn+1 + Fm-1Fn ,  m ≥ 1 , n…

Q: 9. Find a pair of positive integers a<b such that a+b=557685 if possible. Ans:

A:

Q: Show that /(-1)-7(1) but there is Let f(x) 1 Show that /(-1) (1) but there is no number e in (-1, 1)…

A:

Q: 4. Which of the following is NOT true? (a) Ag = Ap:7| ta| Ax (b) А =n Prvn x:n| (c) Ar:ñ| 1 Fn + A 1…

A: Given 4 equations of assurance functions.

Q: (9) -9| <2. ans./ (7, 11)

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

Q: 18. Find Itm,..(xe -x) O 2 O O O

A:

Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

A:

Q: dx_ -In x – 2 |x+2 X - + C x² – 4 - 4 Select one: O True O False

A:

Q: (15 – b) (9 – b) - z = 6. 6. För a = 1 och b + 3 saknar systemet lösningar. För a = 1 eller (4t) , y…

A:

Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: Determine the numbers a such that x a (p) is solvable for p = 7, 11, and 13. %3D

A: Given, x3≡amod p At p=7 We know that suppose p is rational prime p≡1mod 3 then x3≡a mod p is…

Q: - Evaluate the followings: (i)-r' (4) (ii)-(2) (ii) (-3) Г (iv) Γ(0.5)

A:

Q: If In x + In(x – 2) = ln(7x), then x -

A: From the given problem:   lnx+lnx-2=ln7x   Now using the logarithmic property:   ln(a·b)=ln(a)+ln(b)

Q: Find SU U. 10 9. *+6 x+10 S

A: We have to find length of secent US.

Q: Prove that 3! 2(n+3) = = n r=0 (-1) 1 n "Cr r+3 Cr

A:

Q: 4. Show that if x, y, z are integers such that x2 + y2 + z? = 2xyz, then x = y = z = 0. (H)

A:

Q: (1 2 43) 32 1. IF A = THEN (A) (B) (X Y) Z W

A:

Q: 2. How do you prove a statement of the form ( 3x, P(x) )? Provide examples.

A: To prove a statement of the form (∃x P(x) ), we need to find an example value x such that P(x) is…

Q: If X~ Uniform(b, 1) and o² = 3μz, what is b? (a) -0.82 (b) 12.18 (c) 20.82 (d) 17.00 (e) None of the…

A: To find the value of&amp;nbsp;b, we first need to find expressions for the mean μX​ and…

Q: S' = 4 + ( 20 ) * + ( ਲੰਡਨ ) * ( ) 1 + 2022 * 1 + 2021 2 2 + ... + 1+ + 1+ ² 2021 2020 1+ Find S = ?…

A:

Q: 1- 9/if it of Prove that yś(-)- TT

A: In the question it is asked to prove y'(0)=π for the given function y=2xcos-1(2x)-1-4x2.