Prove that n(n + 1)(2n + 1) (3n2 + 3n - 1) 30 whenever n is a positive integer.

Q: Prove that if n is an odd integer, show that n⁴ + 4n² + 11 is expressible in the form 16k for some…

A:

Q: n2 +2 2 3n for n = 1,2,3,.

A:

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: Show that for positive integer n,

A:

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

Q: Use mathematical induction to prove that 1 + 2 + ⋯ + ? =n(n + 1)/2 for each positive integer ?.

A:

Q: Use induction to prove that 3 | (10n+1 + 10n + 1) for all positive integers n

A:

Q: nd the smallest positive integer N such that n+2 3n-4 -}| < < 0.01 for all n ≥ N.

A: We want to find N such that n+23n-1 - 13 &lt; 0.01    for all   n ≥ N

Q: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

Q: Prove that 8 | (54" +7) for every positive integer n.

A:

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

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

A:

Q: Prove that if n is an odd integer then 7n+5 is even.

A: Given Statement: Prove that if n is an odd integer then 7n+5 is even

Q: Consider the following statement. There is an integer n such that 2n² - 5n + 2 is prime. To prove…

A:

Q: Prove {12a + 4b : a and b are integers} = {4c : c is an integer}

A:

Q: Find all positive integers n for which n1º +1 is divisible by 10.

A:

Q: n2 - 2 is

A:

Q: n. Prove that 21|(3n7 + 7n³ + 11n) for all integers

A:

Q: 5. • Use induction to show that for n ≥ 1 we have (3n+ 1) + (3n+4) + (3n+7)++ (6n-2) M n(9n-1) 2

A:

Q: п(п + 1)(2n + 1) 2. Prove that 12 + 22 + 32 + 4?+ ...+n2 for every positive integer n. 6.

A: Here we have to prove that       12+22+32+...+n2 = n(n+1)(2n+1)/6    for every positive integer n .

Q: Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in…

A: Given: Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. Also, the…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.