4b. Show that there exists i and j with i < į such that the consecutive sum p;+pi+1++p; is divisible by 37.

Q: Suppose that E and F are points on the number line. If EF = 20 and Elies at 4, where could F be…

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Q: 4. Let IR be the set of all real numbers, xER and n EN such that n 2 2. If (1+x)> 0, prove that Pn :…

A: Let ℝ be the set of all real numbers,  Let x∈ℝ and n∈ℕ such that n≥2 where ℕ is the set of natural…

Q: If Sn denotes the sum of the first n terms of an AP prove that S30 = 3 (S20-S10)

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Q: If A = (; A-: )-..R. e M2x 2(R), find A" for any positive integer n.

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Q: 2.6.10. Take X = R, and find E°, DE, and E for each of the following sets. (a) E %=D {1, 글, 흙, }.…

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Q: Find the length s of the path (3 + 10t,9 + 2r3/2) over the interval 5 < t < 6. (Express numbers in…

A: The solution are next step..

Q: d) If n iš a If t denotes the nth triangular number, prove that in terms of the binomial…

A: The binomial coefficient, nr is defined using the formula, nr=n!(n-r)! r! . The binomial expression,…

Q: 2. Let, a1 = 3 and for n 2 2, an = 2am-+1, express an in terms of n. 3. Let, a1 =3, a2 = 4 and for n…

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

Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

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Q: For ƒ(x) = sin x, find ƒ"(x), ƒ″(x), ƒ‴(x), and ƒ(4)(x). What is ƒ(4n)(x) when n is a nonnegative…

A: Given  f(x)= sinx Lets take first derivative f'(x)= cos(x)f''(x)= -sin (x)f'''(x)=…

Q: Let bị = 4, b2 = 2, bz = 7, and b4 = -5. Calculate the following sums. (Use symbolic notation and…

A: Given: b1=4,b2=2,b3=7 and b4=-5

Q: 4. Show that for all formulas p, p,0 E S(P), (a) (9 = 4) = ¬(4 ^ ¬ab).

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Q: Find the largest number d such that if |x – 3| < 8, then |5x – 15| < ɛ, where ɛ = 1 Repeat and…

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Q: by xRy if and only if 7 divides x² – yʻ.

A: A relation is equivalence if it is reflexive,symmetric and transitive

Q: Suppose that > a; = -6 and > b; = 15. Compute the %3D i=1 i =1 sum. 111 E (17a; + 116;) i=1

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Q: for all integers n >1 E, i = n(n+1} %3D 2

A: To prove the sum of n natural numbers n≥1 is  ∑i=1ni=n(n+1)2

Q: If m and n are rational numbers such that √m +√n = √5+√24 ,calcule a possible value for m² + n²

A: Given that √m +√n = √5+√24 To calcule a possible value for m² + n²

Q: 5. Find the sum of the first 10 terms of An, where 1, and a2 an 2aп-1 + an-2, а1 %3D %3D = 1.

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Q: M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper…

A: Given:   If M and N are both bounded subsets of R. M+N :=m+n : m∈M  and  n∈N

Q: Use direct proof to show that a. if m and n are odd integers their sum m+n is even. b. If n is odd…

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Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: Suppose that E and F are points on the number line . If EF = 11 and E lies at 3, where could F be…

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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: Given x > 0 and a real number y, there is an n in N such that nx > y.

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Q: So this is the pitcure of the quesiton I did with all the work done. I *think* the area is correct,…

A: We solve this problem by rearranging the triangles in octagon.

Q: 6. Complete the following interval sign table indicating whether the value of each factor and the…

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Q: Use the Geometric Sum identity to prove that, if r 70, then (1 – x)3 1 =1+ (1 - x) + (1 – x)² + |

A: The objectives to use geometry sum identity to prove the following expression.

Q: The value of k such that P2(1) = 2.8 , 2 k 4 1 Is

A: Lagrange's interpolating polynomial is:…

Q: Q2) n(x) = x1 , kx) = (x – 1)/2 , %3D %3D | 1-(8 + x) Find a formula for ( monok).

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Q: 8. ) Find the smallest values of x and y such that 27

A: Given query is to find the smallest value of x and y.

Q: 1. Find the interior and boundary of the following sets: a] A= {1-:n€ N} %3D b] N c] Q d] Úb, 1-]…

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Q: For each positive integer n, show that и(п)и(n + 1)u(п + 2)u(n + 3) — 0

A: To prove : μ(n) μ(n+1) μ(n+2) μ(n+3) = 0  for each positive integer n.

Q: Find the least value of “n" so that , k=n 1 199 Sn 2k-1 k=1 200

A: Given:          Sn=∑k=1n12k−1&gt;199200

Q: 4. Let R be the set of all real numbers, xER and n E N such that n 2 2. If (1+x) > 0, prove that P,…

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Q: For each integer n > 2, let P(n) bè thế för n-1 Eili +2) = n(n-1)(2n + 5) 6. %3D i=1

A: Given                   P(n):   ∑i=1n-1i(i+2)=nn-12n+56,          n≥2

Q: Let M=( 3 Find formulas for the entries of M", where n is a positive integer. Mn = =

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Q: Solve this inequality for n. 5 < 0.00001 0.4 n 5 s 0.00001n04 Multiply both sides by n° 4. 500000 s…

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Q: Which of the following sequeces in GP will have common ratio 3, where n is an Integer? a) gn = 2n2…

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

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Q: HalW.d Solve the following finst order OE.

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Q: Find values of k that satisfy the following equation: JE 1 2 0 1 [1 1 k] 2 3 3 1 k

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Q: (b) Let A = a + 1 and B = b+ 12. Using proof by induction, show that A+B divides An+1 + Bên+1, What…

A: here i used basic condition of divisibility and i used the notation a|b for a divides b .

Q: Find integers s and t so that s· 201 + t · 302 = 1 . S %3D

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Q: a + b (2) (a) Prove by contrapositive : Va, b e R* : a < b → a <• (b) Prove by contradiction : For…

A: Part(a) We need to prove by contrapositive. In proof by contrapositive, if we have a statement "if p…

Q: show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

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Q: Find all m, n E N such that: n! + 2n + 49 = m²

A: This problem can be solved by graphical approach. Use a graphing utility to plot the graph of…

Q: 1. Find the minimum, maximum, supremum, and infimum of the following sets. Justify your answers. (a)…

A: a Let the given set be   n+-1nn;n∈Nminimum of this set is '0'. it obtained by putting n=1 and…

Q: nd the supremum of each of the ecimal place.) , 12, 3} .2

A: Introduction: If there exists a real number M ∈R, called an upper bound of A, such that x∈ M for…