n+1 In+1 = n+2 In-1 + n+2 (n> 1).

Q: 10 +(-1)" · n How many terms are needed to estimate to 5n n=0 within 0.1? Justify your answer!

A:

Q: if 3 does not divide n then 3 divides 2n^2+1

A: Let us Assumue 3 divides (2n2+1) n2+1=3λ Now  2n2+1=n2+1+1⇒(2n2+1)+(-λ)3=1⇒ gcd(2n2+1,3)=1…

Q: n(n+1)(2n+1) Prove: 12 + 22 + 3² + ... +n? for all positive integers, n.

A:

Q: 3-n (a) - 3n n=1

A: According to guidelines we solve one question.thanku

Q: Mathematical Induction to that P(n) is true for all 1. En= 3. (4n-2) n+1) 2. n- 4. (3n-

A:

Q: (a) Ratio Test: (n+1)! n=0 n+2 (b) Ratio Test: 51-n (n+ 1) n=1 5n2 – 2n +1 3n2 +n – 3 (c) Root…

A:

Q: n 3 Prove Σ n i=1 || n² (n+1) ² 4

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Q: Prove by mathematical induction    1+5+9+...+(4n-3)=(2n+1)(n-1) for n≥2

A:

Q: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

A:

Q: K CK-1) In(k-1) k-3

A: To find if ∑k=3∞kk-12lnk-1 is convergent or divergent: ∑k=3∞kk-12lnk-1 is convergent. To show this…

Q: 8. 3n+1 .251-n n#1

A: We have to solve given problem:

Q: 1. Complexity 2n – n³ + 2 Prove that O(n) 3n2 + 4n – 1

A: The big O notation is used to mention the important term r the biggest term in the equation. The…

Q: (b) Use induction and the Product Rule to prove d. -I" = nr"-1 for all n E No.

A: First principle of mathematical Induction : Suppose Pn is any statement dependent on n  where n∈ℕ .…

Q: Using mathematical induction. Prove that Ž {i= n(n+1) WHEN 12 1 2

A:

Q: se mathematical induction to prove that the following statement is true: n(4n+ 1)(4n+5) 3x5+7x9++…

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Q: (a) > [Integral Test] n2 + 108 n=1 1 ( b) Σ (2n + 3)17 [Comparison Test] n=1

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Q: 3 for all n.

A: We have to find the following conditions according to the given information

Q: 2) S(n)-10-2 at n 5

A: Given ,  f(n) = 10 (2)n     ......... (1) We have to find the value at n = 5 Put ,  n = 5 in…

Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n

Q: How many integers n where 1≤n≤1600 satisfy the condition ⌊n⌋\n.

A: Here, need n such that n/⌊n⌋ where 1≤n≤1600 and n is integer. Now, ⌊n⌋ is said to be floor function…

Q: 44-16=n-21 What does n stand for?

A: If, a = x - b. Then, x= a + b.

Q: 00 (2n +1)" n2n n=1

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Q: Show that n 1 i2 i=1 <2 - n

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Q: c) n° n° + 2n is a multiple 3 for all n 2 1.

A: To show: n3+2n is a multiple 3 for all n≥1. Statement used: Principal of Mathematical Induction:…

Q: Induction. 1.3 + 6+9+...+3n= 3n(n+ 1) , for n 1.

A:

Q: Sum (n=0, infinity) (-1)n 22n+1/(2n+1)!

A:

Q: Prove or disprove: 3n = ω(2^n ).

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Q: Prove by the principle of mathematical induction that 3n2 + 3n + 1 < 2n for n ≥ 8 Ensure that you…

A: Given 3n2+3n+1&lt;2n for n≥8

Q: 3 divides n^3 + 2n, whenever n is a positive integer. Solve using mathematical induction

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Q: 1 + 2n < n° for all integers n² 2

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Q: k ak ak a-1,a +1 (-1)* 1/½(-1)*+1 kak 4), a +1 a- (ak + b)(n) (ak+b)(n+1) a(n+1)

A: ∆ak= ak+1-ak      = ak(a-1) therefore anti difference of ak=ak/(a-1)

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: I. Induction a. 1 + 2 + 2² +: · +2" = 2n+1 _ 1, for all nonnegative integers n.

A:

Q: 2. Prove that n²>2n for n >2

A: Note that, the given statement P(n): n2&gt;2n for n≥2 has little correction because if n=2  then…

Q: Σ >k· k! = (n + 1)! – 1 k=1 for every positive integer, n

A: We will prove this by principle of mathematical induction.

Q: Prove using mathematical induction.  1+3+5+⋯+(2n−1)=n^2, where n^2 means "n squared"

A: Principle of Mathematical Induction: Mathematical Induction is a technique of proving a statement,…

Q: (1+i)²n n! n=0 evaluate

A: The function ex has the expansion ex=∑n=0∞xnn!. The complex number eix=cos x+i sin x.  This is…

Q: Prove Γ (n) = (n-1)! where n ∈ ℕ

A: Concept: The proof for this gamma integral will be done using mathematical induction

Q: Show that...

A: We have to show that: n1=nn-1 for n≥1. We know the formula, nr=Crn=n !r ! n-r ! Applying above…

Q: H(n) = 0 if n is  less than or equal to 0  H(n) = 1 if   n=1 or n=2  H(n-1) + H(n-2) - H(n-3)  if…

A: The conditions for H(n) is given as,

Q: using Induction: 2+4+6+··+2n = n(n+1) 8. Prove Mathematical %3D

A:

Q: series: n(2n + 5) 8" (n + 1) In(2n + 3)

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Q: (n+1)! %3D Solve for n: = 30 (n-1)!

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Q: an 1 √2n In n an= n² 2 1+5n4 n=2n4+6n³ an=7

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

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Q: Σ xn + 6 In n n = 2

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Q: (n+1)!–1 Prove that - + n + (n+1)! is true for all 2! 3! (n+1)! integers n > 1.

A: The answer is as follows:

Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…