prove the statement  by mathematical induction.  For any integer n >= 0, 7^n-2^n is divisible by 5

Q: Use mathematical induction to show that 5? − 1 is divisible by 4. Determine the starting value as…

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Q: Suppose you draw n lines in the plane so that every pair of lines cross (no lines are parallel) and…

A: Let n be the number of non parallel lines. Given that number of regions= Rn Let Un represent maximum…

Q: Distinguish between finite and infinite?

A: Finite means that the quantity is limited . It does not mean that it can be calculated on the piece…

Q: -, 1-, 3, 6, 12, 24 ...; find the 10th term and the sum of the first 10 terms.

A: The solution is given as

Q: How many 3-digit positive integers exist that when divided by 7 leave a remainder of 57

A: 3 digit positive integers start from 100 and ends with 999. Least positive 3 digit integer which…

Q: In the expansion of + 6)-", state the number of terms and the coefficient or the middle term. 21, 20…

A: Introduction: The binomial expansion is a very important finite series in mathematics. With the aid…

Q: Find all three-digit numbers with different digits with the property that those numbers are…

A: Given that it is to be found all three-digit numbers with different digits with the property that…

Q: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

A: Let x be the required number. The number is divisible by 3, so it leaves the remainder 0 when it is…

Q: Prove by mathematical induction

A: The principle of mathematical induction is a technique or method that is used to prove the statement…

Q: Give a combinatorial interpretation of the coefficientof x6 in the expansion (1 + x + x2 + x3 +⋯)3.…

A: According to the given information, it is required to give a combinatorial interpretation of the…

Q: n2 1, n - nis divisible by 3.

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Q: When four positive integers are added three at a time the sums are 1551, 1659, 1720 and 1736. The…

A: when four positive integers are added three at a time the sums are 1551, 1659 ,1720 and 1736. we…

Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: Find the Sum to infinity of the sea tès - Ce30 t ces 20 - 1-3 1-3.5 ces 30 + - (-nCBCIT). 2.4 2.4-6

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Q: A number N is divisible by 4 iff the number formed by the last 2 digits of N is divisible by 4. For…

A: We can find number divisible by 4  by using the concept given in the question

Q: 4. One way to add up n natural numbers can be described as: ("+1) which is in 'combinations' form.…

A: Generally sum of n natural numbers by using combinations we can find them using n+1c2. By using this…

Q: find a simple recursive definition of the sequance with general term tn=2(n-1)+7 for n>2

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Q: Determine whether the statement makes sense or does not make sense, and explain your reasoning.

A: Given- One of the term of the binomial expansion is 75x2y4. To determine- whether the statement…

Q: All composite number can represent as product of primes numbers

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Q: Determine the formula and (largest) domain for the following expression and then prove it using…

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Q: Derive an expression for the sum of the first n positive even integers.

A: We Use the A.P series. Find The sum of the first n positive even integer.

Q: A sequence generated by the length 3 recurrence starts 001110. Find the next four elements of the…

A: Given that the sequence is generates by the length 3 recurrence relation. Therefore, the recurrence…

Q: Explain how to expand (a + b)' using the binomial theorem n an-kbk. Be specific. k expression, k-0

A: Given:      a+b7 The binomial theorem:        a+bN=∑n=0NNnaN-nbn where Nn=N!n!N-n!

Q: Find: The sum of all positive integers less than 400 which are not divisible by 5.

A:

Q: What is the 5th term in the expansion of

A: The 5th term of the given expansion is solved by using binomial expansion formula (X+Y)^n.

Q: a sequence generated by the length three recurrence starts 001110 find the next four elements of the…

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Q: The sequence 8, 32, 128, 512, … represents the number of sit-ups Billy did during each workout…

A: Given, 8,32,128,512... Recursive rule means using previous number to get the next sequence number.…

Q: noosing a positive integer from 100 to 999, what is the poSSibility of choosing a positive integer…

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Q: Show that n-lis divisible by 8, if n is an odd positive integer

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Q: Determine the probability that the randomly selected number from the interval [0,1] is a rational…

A: To Determine: The probability that the randomly selected number from the interval [0,1] is a…

Q: Find the sum of the first 100 positive odd integers.

A:

Q: Use mathematical induction to prove: For each natural number n, 6 divides (n3-n). I need to write a…

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Q: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

Q: Use permutations to solve. From a cooler with 9 different kinds of soda, 3 are selected for 3…

A: The formula of permutation is: Prn=n!n-r! This is the required answer.

Q: In two or more complete sentences, explain how to find the third term of the expansion of (2x+y)4.

A:

Q: Determine all natural numbers divisible by 8 whose sum of digits is less than 10 and the product of…

A: Permutation and combination concepts will be used to filter out all the 3 scenarios in getting…

Q: countable in number, such that

A: Given : W=Iα:α∈A is a collection of open intervals, then there exists a sub-collection , say Ii :…

Q: How can the sum of k * (n choose k) = n(2 to the power of n-1) be proven akgebraically ?

A: Step 1: We have to prove that, Now by the binomial theorem, Now, differentiating the both sides of…

Q: Let n be a positive integer. Give a recursive algorithm for computing the sum of the first n…

A: To find - Let n be a positive integer. Give a recursive algorithm for computing the sum of the first…

Q: Use binomial theorum to expand (1 + 1/x)^7

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Q: Using the fast exponentiation method (page 182), determine 535 mod 47. Show your work.

A: We will find out the required value.

Q: (2) Explain why the rational numbers are those having periodic representation or with- out infinite…

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Q: 4 There is an infinite number of ways to represent 1 on the number line. In the space below vrika au…

A: We can express 1 in an infinite number of ways Have to write the four expressions multiplying by 1…

Q: How many twin primes are there where both primes are less than or equal to 31? Number of twin…

A: Twin prime: If there is only one composite number between two prime numbers, then that two numbers…

Q: Find the sum of the first 80 positive even intergers.

A: Sum of the arithmetic progression with for n terms where a1 is the first term and an is the last…

Q: Serial numbers for a product are made using 3 letters followed by 2 numbers. If the letters are…

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Q: Without expanding completely, find the indicated term(s) in the expansion of the expression.

A: Given expression: 3c+c247 Concept used for the general term:…

Q: Find the sum of all Integers between 84 and 719 which are multiple of 5

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Q: What 2 nonnegative numbers with a sum of 23 have the largest possible product?

A: Given information: The two non negative numbers with a sum of 23.  Calculation: Consider x and y are…