Prove that the sum of n squares can be found as follows n(n + 1)(2n + 1) 12 + 22 + 32+..+n? 6

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

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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: Prove the bi-conditional: prove that if 5n+2 is even, then n is even, where n is an integer.

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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: Prove that the sum 2 + 22 + 23 + 24 + ...+ 22019 + 22020 is divisible by 30.

A: Hi there! Thank you for the question, but as per our guidelines we can solve only one question at a…

Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

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Q: ^n for all n>=6

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Q: Prove that n! > 2" for n > 4.

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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: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: Use mathematical induction to prove that the sum of the squares of the first n natural numbers : Ek?…

A: Let's prove.

Q: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

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

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Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: 2. Find the sum of all rectangles shown in figure below in terms of x and y. x/2 3/4 y/4 y/2

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Q: Prove that the sum 2 + 22 + 23 + 24 + · . . + 22019 + 22020 is divisible by 30.

A: The sum of the geometric series a, ar, ar2, ar3,…, arn-1 is given as the following formula.…

Q: Prove that the sum of the 1st natural n numbers = n*(n+1)/2

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Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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Q: Prove that n! > 2" for n 2 4.

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Q: n(2n – 1)(2n+1) 12+ 32 + 52 + ...+ (2n–1)² 3

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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: 7. а) Prove that (ӑ — Б) х (ӑ+Б) %3 2ӑ х ѣ.

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Q: Find all the odd positive integers n which satisfy ó(n) = 20. %3D

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Q: . Prove that 2 +4+6+•…+ 2n = n(n + 1) for all n > 1.

A: According to the given information it is required to prove

Q: IM: Prove that n(n+1)(2n + 1) %3D 6.

A: We prove the given statement by using the sum of first n natural numbers.

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: Can you assist  me in  verifing    that    for    all    n ≥ 1,    the    sum    of    the    first…

A: Here, we have to verify that ∀n≥1, the sum of the squares of the first 2n positive integers is given…

Q: If n>1 prove the sum %3D is not an integer. 1无

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Q: Prove using Mathematical Induction that: The sum of the squares of the first n positive integers is…

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Q: Show that 2n+1≤2^n  for all n≥3.

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Q: 7 45 and fourth term of (3x -) 6. Find the product of expressed 2 as P.

A: The complete solutions are given below

Q: 2. Prove that n²>2n for n >2

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

Q: 2. Find the sum of all rectangles shown in figure below in terms of x and y. x x/2 x/4 y/4 y/2

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Q: Show that the sum of the squares of the first n natural numbers is given by,S=(n(n+1)(2n+1))/6

A: We have to show that the sum of the squares of the first n natural numbers is given by S=nn+12n+16…

Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: Prove using Mathematical Induction that: The sum of the squares of the first n positive integers is…

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Q: Prove n 2 2n 1 k = n k - 1 n k=1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

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: 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: (c) Express the sums in closed form n-1 3k and k=1

A: Given, ∑k=1n3kn    and  ∑k=1nk3n2   We have to evaluate the sum

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

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Q: 2. Find the sum of all rectangles shown in figure below in temms of x and y. ¥/2 /4 Y/4 Y/2 ere to…

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Q: 10. Prove that the sum of n squares can be found as follows n(n + 1)(2n + 1) 12 + 22 + 32+...+n² : 6

A: I use  a3-b3= (a-b)(a2+ab+b2)

Q: Prove that the sum of the first n natural numbers is given by this formula: n(n + 1) 1+2 +3 +. ..…

A: As per our guidelines, we are supposed to answer only one question. Kindly repost other questions as…

Q: Find the sum of all natural numbers n such that 2021-n/99 is a natural number ?

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Q: 10. Prove that the sum of n squares can be found as follows 12 + 22 + 3°+...+n² n(n + 1)(2n + 1) 6.

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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: for n greater or equal to 1  proove that 2+4+6+...+2n=n^2+n by using mathematical induction

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Q: Prove that the sum of n squares can be found as follows n(n + 1)(2n + 1) 1? + 2° + 3°+.. +n? . 6.…

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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…