Show by induction that for each positive integer k, 1?+ 2² + 3² + . . · +k² = }k(k+ 1)(2k + 1).

Q: Can you use mathematical induction to show that 4 divides 32x + 7 First define the predicate ?(x)

A: given , expression :- 32x+7 we have to prove that given expression is divisible by 4 .

Q: Show that for positive integer n,

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Q: k+4 n(3n +7) k(k +1)(k + 2) 2(n+1)(n+2) k=1

A: Given ∑k=1nk+4k(k+1)(k+2) To show  ∑k=1nk+4k(k+1)(k+2)=n(3n+7)2(n+1)(n+2)

Q: Can you use mathematical induction to show that 4 divides 32x + 7 First define the predicate P(x)

A: Yes we can use mathematical induction for Natural number x.

Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: at any positive odd integer is of the

A: Here we have to show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,…

Q: How do I prove that 4 divides 32n-1+1 with induction? (n is natural numbers)

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Q: Using Fermat’s little theorem to show that 42 | n7 −n, where n is a positive integer.

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Q: Given any real numbers a and d, consider the expression (a + md) + (a + (m + 1)d) + (a + (m + 2)d) +…

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Q: Find the product of all positive integers from n down through 1 for n = 5.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Show that is composite for all primes p. p+14p²-8

A: The composite primes are those numbers which are generated by multiplying the two smallest positive…

Q: prove that the sum of three odd integers is odd using indirect method. show your solution and…

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Q: Prove that √1+i√3+√1-i√3 = √6

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Q: Prove or disprove: There exists an integer a such that 14 | (3a – 5) and 21 | (2a +3).

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Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's states that if p is a prime and a is a positive integer not divisible by p, than ap-1≡1 mod…

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

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Q: п(п + 1) Show that , k 2 k=1

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Q: Find all sollutions in integers of 34x+8y=6

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Q: AV Prove that every integer greater than 6 can be written as 3x + 4y for nonnegative integers x and…

A: We have to prove that:Every integer greater than 6 can be written as 3x+4y for nonnegative integers…

Q: Show that n! > 3ª for n > 7.

A: The given inequality is, n!>3n We need to prove that, the inequality is true for n≥7 Firstly,…

Q: The binomial theorem states that for any real numbers a and b, (a +b)" = E o (") a"-* for any…

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Q: Show that 1+5+9+...+ (4n-3) = 2n²n using mathematical. induction.

A: Base case:    For n = 1:          LHS = 1          RHS = 2.12 - 1 = 2 - 1 = 1      So, LHS = RHS.…