n = n and n = 1. n Show that in

Q: We can use sigma notation to rewrite the sum 4 + 7+ 10 + 13 + + 31 as: ... 4 +7+ 10 + 13 + ..+ 31 =…

A:

Q: 1+i Find the least positive value of n, if =1. 1-i

A: We know that i2  = -1   i4  = 1

Q: Suppose r is a non-negative real number and n is a natural number. Use the Binomial Theorem to prove…

A: The binomial theorem states that for the positive integer value n: x+yn=∑k=0nCknxk+yn-k

Q: 5. Give a general form for T(n) for n > 2

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Q: (d) Given that n is even, how many 00-free bitstring of length n where n is even have exactly (끌 +…

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Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: Show Hhat the series Canverges to 1 Kei (K+)!

A: Introduction: The ratio test is one of the basic tests to verify the convergence of a given series.…

Q: The n-th random number is generated by the function below. If xO=2, x2 is * Xn = (7Xn-1 + 9) mod 11…

A: Given Xn = ( 7 Xn-1 + 9) mod 11 X0 = 2 X2 = ?

Q: arctan(n) (iv) (-1)". n3 +n +1 n=1 [ Select ] n! 5n n=1

A: Solve the 4 and 5

Q: 1 Answer: diverges n In n n=2

A: If the integral ∫1∞fxdx=∞ then ∑i=1∞an diverges.

Q: 2 3 n a. If n E N, then + + + •+ 3! 4! (n+1)! (Hint: 3! (reads 3 factorial) = 3 x 2 x 1. n! = nx…

A: Induction theory: Let the given statement is P(n). Step-1: We need to check P(n) is true for n=1.(or…

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

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Q: 3. Let A={n €Z, 2Z). Prove or disprove: For every n E A we have n² = 4.

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Q: Assume C and D are chosen at random from {1, 2 . . . , n}. Let pn be the probability that C + D is a…

A: Given: C and D are chosen at random from {1, 2 . . . , n}. Then the number of pairs (c, d) with sum…

Q: Σ (-1) n=1 nyerges for all n20

A: Please refer to the image below

Q: Q: Apply Taylov's expansion and find the seyies for tenn at 4

A: Taylor series of a function is given by: ∑n=0∞f(n)(a).(x-a)nn!

Q: 1 2. Let e =. Determine the smallest choice of N > 0 such that N. 6 ' n(n + 1)

A: To determine the smallest choice.

Q: 1. Prove that if n E N, then E, k(k + 1) = n(n+1)(n+2). 3

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Q: verges absolutely or diverges. A 2" 1. n! 00 n=1

A:

Q: 7. Find a number t such that In(4t +3) = 5 (Give EXACT value)

A: Given that ln(4t+3)=5 we need to find "t"

Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: the series 4(4)-8(6)+16(8)-32(10)+64(14)+256(16)-512(18) is represented by which sigma notation

A: Let's find.

Q: all n N we have We apply this successively for n = bn+1 bn N, N+1, n 9 bN+1 < (m + €)bN bN+2 < (m+…

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Q: 27 en+1 – 1 < 0.05 ? 2 (n + 1) · (n + 1)!

A: Given: The expression is, 272en+1-1n+1n+1!. Calculation: Obtain the value n such that…

Q: Compute |A| for the following:

A: Given, we have to find the determinant of A

Q: a 1 In parts (a) - (c), prove that for all n E N: (a) n-1 k k=0 (b) ¿(:)-- 2n+1 k k-0 (c) E(-1)* k=0…

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Q: 7. (a) A computer is programmed to choose an integer between 1 and 99, inclusive, so that the…

A:

Q: Write all elements of S4. Show that S4 has no elements of order > 5.

A: The given question is related with abstract algebra. We have to write all the elements of S4 . Also,…

Q: = (-1)n n. Find a87 Let an

A: power of negative number

Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

A:

Q: The sequence (sn) is defined recursively by s­­­0=3 and sk=sk-1+2k. Show that sn=3+n(n+1) using…

A: We are going use method of mathematical induction to prove the result

Q: Find the largest value of n such that: Σ E (9k – 5) < 3000 k=1 n =

A: We expand the sum and use the know formulas of "sum of first n integers"

Q: 4 + In(n*) \ D) { { } [ Select ] 2 + In n 4 + In(n*) E) n + In n [ Select ] 2n? + 2n + 4 F) { (-1)".…

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Q: Use any method to deter (a) 13 The series 8.

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Q: 8. If n+1, 2n-1 and n+5 are in AP. Find the value of n

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Q: (c) Find the term independent of x in the expansion of, 15 2 +

A:

Q: Sigma(n=1 to n= infinity) 5* (2n-

A: We have to solve given problem:

Q: 1. 4 + 10 + 18 +..... + n (n + 3) = [n (n + 1) (n +5)] / 3 for n e Z+ Use Mathematical Induction

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Q: A natural number n is even if n = 2k for some kw. If n = 2 i + 1 for some i Ew, then n is odd. Prove…

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Q: 1. Show that the binomial coefficient 2"Cn is always an even number for all n > 0.

A: Here, the given binomial coefficient is: Cn2n To prove that it is always an even number for all…

Q: Consider a population that grows according to the resursive rule Pn=Pn-1+95, with initial…

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Q: Which of the following express 1 +2+4+ 8+ 16 + 32 in sigma notation?

A: Option (d) all of them we will verify as below

Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

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Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: Show that for all n € N, one has 1 2√n+1 <√n+1-√n<; 2√ñ

A:

Q: Use Gauss’s trick to determine H(n) = 1 + 5 + 9 + 13 + … + (4n – 7) + (4n – 3) + (4n + 1) I'm…

A: Explained below

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

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Q: Suppose n e z. If n² is odd, then n is odd.

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Q: Find an explicit formula for an if a0= 1 and an= (n + 1)an−1+ 3^n for n ≥ 1

A: Answer