Prove n 2 2n 1 k = n k - 1 n k=1

Q: O Prove 1+3+ 9+27+ ·+ 3n-1 = 3"-1 for any positive integer n. %3D

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Q: 9. Prove the formula n(n+1) >i =1+2+...+n= 2 i=1

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

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Q: („*.) (* ) =(). 2n 2n 2n + 2 (c) Verify that + n +1 n + 1

A: According to the guidelines we are supposed to do one problem

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

A: Consider the provided question,

Q: %3D k (n+1) (k-10 (n+1) k +1 (*) n+1)

A: The correct option is the last option. The detailed solution is as follows below:

Q: For all integers n > 1, E(4i – 3) = 2n² – n i=1 [ In other words: For all integers n > 1, 1+5+9+ ..+…

A: The given series is ∑i=1n4i-3=2n2-n, n≥1 To prove the statement by mathematical induction, we will…

Q: n°(n+1)² 1 +2° +... + n³ %3D for every natural number n. 4

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

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Q: η Prove Σ η i=1 3 = n(n+1) 4 2

A: Introduction: When we prove a statement that is valid for any subset of the set of natural numbers,…

Q: Prove that n/n2 +1 is greater than (n+1)/ (n+1)2 + 1

A: If An and Bn be two term. To show that An is greater than Bn it is sufficient to show that…

Q: (b) Show that k-1 (k+1) n n+1 for every n ≥ 1.

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Q: 1 Prove 2.5 1 1 + .…+ 8.11 1 for n > 1. %3D 5.8 (3n-1)(3n+2)

A: Prove S(n) for n=1.   LHS =1/2.5 =1/10; RHS = 1/6+4=1/10   So it is true for n=1.

Q: (-1)" /n + I is n + 2 n=1

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Q: IM- Prove that n(n+ 1)(2n + 1)

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Q: n-1 For all n > 2, 4+ 3n4" – 4n+1 - j=1 9.

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Q: 1 Let n = Find n + 1 and n + 1 n + 1 = %3D 1 %3D n + 1

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Q: *prove that 1P4 ľ +3'+ ... +n² = n' (n+1)* 4 for n= l,2,3

A: The given expression is the series expression. Formula to find the sum of series helps to prove the…

Q: (-1)"(n + 2) 2n+1(n + 1) 4. Compute the exact value of n=0

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Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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Q: Prove 1 0, Ø ! n n=1

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Q: Prove (6" -1) is ÷ by 5, Vne N 1. Show true for n =1 6' -1=5 2. Assume true for n = k 6* -1 is ÷ by…

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Q: Prove that: n(n+ 1) r = 2 i=1

A: ∑i=1nr=1+2+3+....................+nhere it is a arithmatic progression sum of n terms of arithmatic…

Q: 3(3 + 1) n = 3: Show that 1 + 2 + 3 =

A: To show that 1 + 2 + 3 + ...  + n = n(n + 1)2 is true for the given value of n.   Also, to show…

Q: (-1)"(п + 2) 2n+1(n +1) Compute the exact value of n=0

A: The geometric series formula is ∑n=0∞rn=11-r, where r<1. Simplify the numerator of the given…

Q: 1 + 2n < n° for all integers n² 2

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Q: 7. For every integer n z 1, n(5n – 3) 1+6+11+16++(5n– 4) 2

A: Solution is in step 2

Q: (-1)"(n+ 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A: Given series is  ∑n=0∞-1nn+22n+1n+1

Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: (-1)"(n + 2) 2n+1(n + 1) Compute the exact value of n=0

A: The ratio test can be used to assess if a series is converging or diverging. The following are the…

Q: Prove E-1(4i – 3) = n(2n – 1) for natural numbers n 2 1. i=1

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Q: Prove by Mathematical Induction. n(n + 1)(2n + 1) k² 6 k=1

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Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

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

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

Q: Solve for n if nEN: (4+8+12+ ... +4(n-1)) + (2+4.5+7+. .. 4+5n -%3D126.5 2

A: Sum of n natural numbers is n(n+1)/2.

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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Q: n(n – 1) is O(n²). 2 Show that

A: f(n) is O(g(n)) if there exist positive constants c and k such that f(n)≤c g(n) whenever n&gt;k.…

Q: Assume that n ∈ N. Prove that if 3|(2n + 1) then 9|(10n^ 2 + n − 2).

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Q: n* + n² + 1 4. Prove that O(n*)

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Q: Prove that n^2+5 >=4n for all integers, n>0.

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Q: ) 1+3+5+ ...+ (2n – 1) = n² for all n > 1.

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Q: Find the possible values of n. (n2 )" – 2n" + 1 = 0 (n +0,1)

A: We have  n2n-2nn+1=0 and n≠0,1 We can write n2n as n2n=n2n=nn2 So, nn2-2nn+1=0 Let nn=x x2-2x+1=0

Q: Find σ(n) if n = 2^k-1 (2^k - 3) where 2^k - 3 is prime

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Q: η Σ (1) 5 k=1 k odd = 6" - (-4)" 2

A: Here we prove that ∑k=1k oddnnk5k=6n--4n2

Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

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Q: 1 Let n = Find n + 1 and %3D n + 1 n + 1 = %3D 1 %3D n + 1

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Q: Prove that 2 + 2 · 22 + 3 · 23 + 4 · 2* + ...+ n2" = (n – 1)2n+1 + 2 for all n 2 1. Prove that 12 –…

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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…