Show that a fraction r = a/ b in lowest terms has a finite decimalexpansion if and only ifb = 2n5m for some n,m > 0

Q: Solve for r: In(x) + In(x + 2) = 3

A: Use the properties of lograthim and simplify.

Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

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

A:

Q: 1. Assume that cin. 8" + c2n · 4" + c3n° . 9" = 0 %3D for all n> 0. Show that then ci = 0.

A: We need to the equation c1n5·8n+c2n·4n+c3n3·9n=0 for all n>0 for the value of c1. The equation…

Q: 4.1c. Show that the product of two integers that can be represented by x2 +dy?, is always another…

A: Prove that the product of two integers of the form x2+dy2 is always of the form x2+dy2. Consider two…

Q: Show that 1/2 ( Tn + Mn ) = T 2n

A:

Q: 2. Let n be a positive integer. Show that 4n 4n-k (4n 3 4n k=2n (Hint: an argument similar to the…

A: Solve the question by first opening the summation on the left side of the inequality. Simplify the…

Q: Express In V16 in terms of In 2 and/or In In 16 =[ (Type an exact answer.)

A:

Q: n Prove that + 2 + 22 +...+ 2" +...+ 2n = 3" for all integers n 2 0. * ^) xn-ryr and x=4=1.…

A: We have to prove given identity:

Q: Prove that for all n E N, 1 1x 2 x 3 n(n + 3) 4(n + 1){n + 2 1 1 2 x 3 x 4 n(n + 1)(n+2)

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Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

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: If (ar + br) = 2n %3! n+1 and k-1 8. E(ak - br) = n+2 - , then ar %3D k-1 k-1

A:

Q: of Voots of the equation. 2x+3X+5=0 is The muttiple is Ę 2. 3 Prove That The sum

A: Use the formula for the sum and product of the roots of a quadratic equation.

Q: Show that >n"x" diverges for all x + 0. n=0

A:

Q: Suppose the interval [1,9] is divided inton subintervals of equal width, and C represents the left…

A:

Q: 1. • If r + 1| < o, prove that |r³ + 1| < 0.331.

A: Here we use the fact  | x |&lt; a implies  -a&lt; x &lt; a .

Q: prove that l+2 +3+.. +h = n(ntl) for n=1,2g3 2 %3D

A: Given :               1 + 2 + 3 + ... + n = n(n+1)2 Let P(n) :               P(n) = 1 + 2 + 3 + ...…

Q: - such that n >r > 1, (n – 1)! r!(n – 1 – r) n! r!(n – r)! n - r -

A:

Q: 4. Prove that 3i² + i = n(n+ 1)?

A:

Q: (b) Prove that for any positive integer n, Ej. 2 = (n – 1)2"+1 + 2 (c) n(n² – 1) Prove that for any…

A: b) Let the statement be P(n): 1·21+2·22+3·23+⋯ +n·2n(, or,∑j=1nj·2j )=(n-1)2n+1+2. For n=1, P(1) :…

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: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

A:

Q: Prove that lor n {k=1 positive unteger every K2K = (n-1)2²+¹+2

A: answer is in next step.

Q: The interval [0,5] is partitioned into nn equal subintervals, and a number xi is arbitrarily chosen…

A: The given interval is [0,5]. Find the xi.

Q: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

Q: *3. Prove that l' + 2 + .. +n =n(n+ 1(2n + 1) for all ne N.

A: By using induction.

Q: Find x for which det(A)=0 7-x -2 A = 1- x 0. 5- x The sum of the answers is....

A: See the detailed solution below.

Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

A:

Q: The average pollen count in a certain city on day x of the pollen season is  P(x) = 4x − 0.2x2 for…

A: The average pollen count in a certain city on day x of the pollen season is P(x)=4x-0.2x2…

Q: 2. (a) For n ≥ 0 and x E R, prove that (b) Prove n n Σ ^ (^.) 2^* (1 - 2)"-* k k=0 = = In. n x)n-k…

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Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

A:

Q: 5 = t(n + 1) = t(n +2) = t(n + 3) holds for n = 3655 and 4503. (a) Verify that t (n) (b) When n =…

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Q: Forthe Benstein felynominte prove that こニ-x)() c} ヒ。 4n

A:

Q: Prove that 2+1> (n + 2) · sin(n) for all positive integers n.

A: Concept: Principle of mathematical induction: For each natural number n, mathematical induction is a…

Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

A:

Q: Suppose x, n ∈ N and n not equal to  0. Then, show that xn is positive if and only if x is positive

A: Given that x, n∈N and n≠0. According to question, N=0,1,2,3,4,5........ First suppose xn is positive…

Q: part ii and iii, thank you

A: According to pascal law,

Q: Suppose m=nd where n is odd and 1<n<m. Prove that 2m+1 is composite.

A:

Q: Let Hn = the sum from i= 1 to n of 1/ifor any n ≥ 1. Prove or disprove that for any n ≥ 1H2n ≥n/4.

A: So, Hn looks like,             We will use mathematical induction to show that,…

Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: What are the value for sup e-* = е x>0 and inf e * - = XER

A:

Q: Check that o(-2)" (":) = 1– 2(;) + 4(;) + ... + (-2)" (") + ... + (-2)" in ir=0 cannot be greater…

A: Given sum is  ∑r=0n-2rnr=1-2n1+4n2+...+-2rnr+..+-2nWe need to prove that the given sum cannot be…

Q: 2n° 1. Prove that for all positive integers n > 1, the inequality 2" < <(") holds. п

A: Consider the given statement be P(n) as; Apply induction methods to prove the above statement P(n)…

Q: Suppose the interval [1,9] is divided into n subintervals of equal width, and Ck represents the left…

A:

Q: 4. Prove that the L{2t²} = 4 S3

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Q: 4. Prove that n(3n – 1)r x + 4x + 7x + ... + (3п - 2)x for n e N.

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Q: Let f(n) = 2n – n? + 10n – 7. Show that f(n) is O(n³) using specific values of C and ng-

A:

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.