2(5) – 3(1 – 6) Is even integer?

Q: 16. The expansion of (2 + 3x;-1 is valid when >x>

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Q: 2x+ 3 2 X 6

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Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

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Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

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

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Q: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

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

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Q: Prove that 2" < n! for any integern 2 4

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Show that2/31 + 2/32 + g+ 2/3n = 1 - 1/3nfor all positive integers n.

A: The given series is the geometric series, hence the formula to find the sum of the series helps to…

Q: If x is an odd integer then 5x – 7 is an even integer.

A: 5 and x are odd integers. Multiplication of two odd integers is always odd. Hence, 5x is an odd…

Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: Suppose, and no are the smallest of two positive integers so that 300x is a perfect square and 300y…

A: The given problem is to find the values of x and y and sum of value of x+y for which smallest…

Q: Show that (n + 1)!)" < 2! · 4! - (2n)!.

A: The objective is to prove the given statement using induction

Q: How many terms are there in the expansion of (4.x + 7y)0.+ (4x- 7y)10 ?

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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: What is the coefficient of the term containing x^4 in the expansion (x - (2/x^2)^10?

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Q: What is the fifth term of the expansion of (x^2 + 1/x)^10?

A: What is the fifth term of the expansion of (x^2 + 1/x)^10?

Q: If n is an even integer then 7- n3 ) is an odd integer. ) is an even integer. is sometimes even and…

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Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

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Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: Prove that 12 - 2²+ 32 - ... .. + (-1)a-1n²=(-1)n-1n(n+1)/2

A: The given problem can be solved using mathematical induction:

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: Show that yn for all n=1, 2₁- ✓

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Q: Prove 3√3  is irrational

A: See the attachment.

Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: Show that 2n+1≤2^n  for all n≥3.

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

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Q: prove that √2+√3 is irrational

A: For rational we needed form p by q P and q are integer

Q: 4) Find the Term that free from x of the expansion (x2 +10

A: Use binomial expansion:-

Q: 3. Use the binomial theorem to prove that 2n 2n |22n–2k = (2.5)2". k k=0

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Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

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Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: 10 What is the 6th term of (x − 1)¹⁰-

A: To find 6th term of x-110. Formula for general term of expansion of a+bn is given as -…

Q: Show that 2^n &gt;n^2 if n is large enough.

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Q: Use a combinatorial argument to show that n 1(n 3 |

A: The LHS of the expression is nn-2. This is the number of ways selecting n-2 objects from n objects.…

Q: Prove that 3/3 is irrational.

A: A number which can be written in the ratio , where and are integers are called rational numbers.…

Q: Prove that (3+2/5) is irrational

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Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

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Q: 4. Use a combinatorial argument to show that () n 1 -2) n + 22 n 2 3 -

A: Given: nn-2=n3+12n2n-2n

Q: Prove that 3+√3 is irrational.

A: As per guidelines we are supposed to solve the maximum one questions at  a time kindly post another…

Q: Prove that 2+/3 is irrational

A: Given,          To Prove : 2+3 is irrational.

Q: 1+ 3k 1- 3k] 1 – 3k 1+3k 2 - prove that AK: 2 1 For A : -1 -

A: Given, A=2-1-12   ...i    ....(i)  We need to show that Ak=121+3k1+3k1+3k1+3k    ...ii     Now, A…

Q: What is the coefficient of z^4 in the expansion of(z^3+3z+1)^5?

A:

Q: 2. Prove that if 2n² + 7n is even and n is an integer, then n is even.

A: We have to prove that the given statement is true by the help of mathematical induction

Q: 2. 6n – 1 is divisible by 5 for n any positive integer

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Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: Obtain the expansion of : 2 up to the term in x8 1-3x2

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Q: Question 8: Using combinatorial argument, prove that (:) --C) - »C) * 311 +n* 3

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Q: Use a combinatorial argument to show that n 1 2 - 2 3 2 2

A: To use combinatorial argument to prove whether the given statement is correct or incorrect. The…