. Use mathematical induction to show that n Σ r=1 (2r − 1)² = 1² + 3² +5² + + (2n − 1)² = = n(2n − 1) (2n + 1) 3

Q: Show that the middle term in the expansion of (1 + x)2¹ is (1-3.5 (2n- -1)) 2"x", where n is a…

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Q: P(X₁ = x₁, ..., Xn = xn) = P(X₁ = x₁) [I²²2 P(X; = Xi| X₁ = X₁, ..., Xi−1 = Xi−1 )

A: Here we have a given equation that

Q: 1+ x Σ Aa" (1 – 2)? n=0

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Q: Find the Cholesky factorization A-[-+-+4] -1 20) A =

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Q: 1. Prove the following equality by induction: 1- 22 + 32 – 4? +. + (2n - 1)? - (2n)² = -n(2n + 1),…

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Q: (2n)! – (2n + 1)! (2n)!

A: It is required to find out , 2n! - 2n+1!2n!  For this we use factorial properties.

Q: 2. Using the mathematical induction, prove Σ (k² + 2k-2) = n(2n² + 9n - 5), k=1

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Q: Expand and simplify. (8h +3) (7h² – 4h+1) - O 56h3 – 32h² + 8h + 3 - O 56h3 – 11h? 4h + 3 O 56h3 +…

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Q: State whether the following is true or false: g) The set of integers the usual metric is not…

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Q: Differentiate f (x) = 71n(x²-6x+3) A. (2x – 6) 7ln(x²-6x+3) f' (x) = x2 – 6x + 3 - В. In(7) (2x – 6)…

A: Given query is to find the derivative of the expression.

Q: Q:- Show thet B(n, n) B( n+on+=2 1-40

A: The solution is given as

Q: Prove that: n² + 3n 1 →> 2n² + 1 2 You may use any results from lectures, but you must state them…

A: (.) Given,                We find the limit.

Q: a. Is (x - 1) a factor of x" c. Is (x - 1) a factor of x" - 1? e. If n is a positive even integer,…

A: Hey, since there are multiple sub-parts posted, we will answer first three sub-parts. If you want…

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: TRUE or FALSE. The one-form c = (x + e + 2xy) dx + (y + y²) dy on R² is closed. Select one: True…

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Q: Find the coefficients of a* in (6 – ½a)6 and kª in (3k – 21)¹2

A: We have to find the coefficients of a4 in 6 - 12a6 and k4 in 3k - 2l12 . Note that,       a - bn…

Q: 1. Prove by induction that for all n € N>1 we have (a) (b) (c) 2 +3 2.3 9+ (9 × 10) + (9 × 100) ++…

A: (a) Given 2n+3<2n·3 For n=1 L.H.S=21+3=5 R.H.S=21·3=6 Thus the inequality hold for n=1 Let…

Q: Let n eN 1. Show that o (n²) = ng(n) 2. Show that o (n) +

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Q: What is the rule for this pattern? (figure 1 is 1 block) Of(x)=n+2 Of(x) = 2n Of(x)=n-1 Of(x) = 2n-1

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Q: 13 - What is Taylor polynomial with order 3 generated by f(x) = x° + 3x² + 2x + 4 at x = 1.

A: The Taylor polynomial of order 3 is given as follows :

Q: 2. Expand the following summations: a. b. Σxy c. Ef-o(2 + z;) d. E- x+ y e. E(axi+ b)? f. E(2xi+…

A: Multipart:  We need to expand given summations ,  We know that , summation is the addition of…

Q: 1 Σ (-3)²+" (n² + 1) -(4x – 12)" n n=0

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Q: . Use Mathematical Induction to show that for all n > 1. 71 i=1 F2i = F2n11-1

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Q: resubmit

A: We know about 3 important coordinate System which are Cartesian Coordinate System represented as…

Q: Prove the following fact by induction: n-1 For all n > 2, j. 4i 4+ 3n4" 4n+1 j=1 9.

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Q: Let (M) be the statement P(n), n 2 1 where P(n) is 2[1+3+32 + ·.+ 3"] = 3n+1 – 1. (M) is a false…

A: For n=1, L.H.S. = 2[1]=2.

Q: (A) I = and (B )I = . Find 2A D - BD O O O none of the above

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Q: hich of the following expressions are equivalent to 2n 4n2 + 4n n+ 5 4n2 + 6n 10 (n+ 1) C 4n (n+1)…

A: given,

Q: Σ s2 an convergent or divergent?

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Q: 2n+2 – 3" Σ - 2. Determine 6n+1 n=0

A: To find: ∑n=0∞2n+2-3n6n+1

Q: Select all of the following that are equivalent to >(2) +> (3k + 1)? k=0 k=0 п+1 A. O) (3k + 3) k=0…

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Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

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Q: (i) in n(n+ 1) r = r=1 (ii) n(n+ 1) (2n + 1) 6. r=1 (iii) in 2. n(n+1)]2 3

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Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

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