3(3 + 1) n = 3: Show that 1 + 2 + 3 = show that n(n + 1) 1 + 2 + 3 + ·.+ n = 2 is true for the given value of n.

Q: Prove that if n is a multiple of 4, then Fn is a multiple of 3.  

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Q: Show that if x > 9, then x³ – 8x? + 2x > 3. Show that for all n e N, if n > 10, then nª – 2n² > 3n³.

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

A: Consider the provided question,

Q: n! %3D if n is non-negative integer and proof r-1 r>t

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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: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

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Q: The statement 2n< n! is valid for For only n=4 All positive integers All positive integers n 2 4 All…

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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: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

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

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Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: The small positive integer n for which (1+i)?"=(1-i)?n is =..

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

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

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n&gt;=1 in question number (7) Sum of squares of first n natural numbers…

Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: Show that if the statement 1 + 2 + 22 + · · · + 2n - 1 = 2n is assumed to be true for n = k, then it…

A: Given, 1 + 2 + 22 + · · · + 2n - 1 = 2n ------------(1) is assumed to be true for n = k To prove -…

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: Show that n*-n3.5+300n + 234n²log n³ + nVn10 e 0(n*)

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Q: Prove that n3 + 2n is divisible by 3 for all n E N.

A: We prove that n3+2n is divisible by 3 for all natural numbers n.

Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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Q: For how many positive integers 'n', n3-8n2 + 20n - 13 is a prime?

A: Introduction :

Q: 3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d |…

A: To solve the given questions we will use the following formulas : 1.   We say that two positive…

Q: For all integers n> 0, 32n – 1 is divisible by 8.

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Q: (-1)*-1 '(nIn n)? n=1

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Q: hich of the following expressions are equivalent to 2n 4n2 + 4n n+ 5 4n2 + 6n 10 (n+ 1) C 4n (n+1)…

A: given,

Q: 5. Find and prove a formula for P(n,n-2) for n2 3.

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

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Q: Prove the statement for positive integer n and r, with r ≤ n. P(n, 1) = n

A: To prove the statement

Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

Q: (4) For all natural numbers n, 4^n - 1 is a multiple of 3

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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: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

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Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

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Q: [Hint: For m > 2, ( 2 7. For n > 1, verify that + 1 1² + 3² + 5² + ... + (2n – 1)° = ( 2"" ) - 8.…

A: Let's say we need to prove a formula or a statement P(n). To prove this, we need to prove that The…

Q: 6. Show directly that inf(B) = 1 for B {N =: 4+1} -

A: The given expression is,  B = {1+1/n : n belong to N}

Q: 5. Prove that the following are irrational: E-2-3', £-12¬!.

A: Expand the given functions and then determine their values. If the values are irrational then the…

Q: teger n is odd if and only if 3n + 1 is even.

A: Here I am providing my handwritten solution

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: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: Prove that 4 + 8 + 12 + .... + 4n = 2n (n + 1 ) Please write it clearly

A: Solution : We have given the expression 4 + 8 + 12 + … + 4n = 2n(n + 1) and We need to prove the…

Q: Prove that 2^? +3^? is a multiple of 5 for all odd ?∈N.

A: We will solve this problem using principle of mathematical induction.

Q: Given: 21 = 2 23 = L4 e || n Prove: 2 3

A: …

Q: What is the second step to prove that S, = 2+ 22 + 2' +...+ 2* = 2(2* – 1)? Show that S, is valid…

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Q: . Let, a, = 1, az = 2, b; = 0, b2 = 1, for n2 3, a, = 2an-1 + bn-1 for n22 b, = br-1+ an-1 express…

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

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Q: Suppose we prove that 2" > n? for n=1 and 2. Knowing that the inequality is true for some n n > 4…

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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…