Use induction to prove that 5^(2n) − 4^n is divisible by 7 for all natural numbers n.

Q: Expand (7 – 3x)“ in ascending powers of x, up to and including the 4" term. Hence, approximate…

A: We know (n+1)! = (n+1) n! Crn = n!(n-r)! r!

Q: TION 9 Prove by Induction that for every natural number n, 3"–1 is a multiple of 2? Attach File…

A: We prove by induction

Q: Use a direct proof to show that the sum of two odd integers is even.

A: Types of integers : Odd integers : The integer which is not divided by 2 . It has the form 2k+1…

Q: Show that there are infinitely many primes.

A:

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: 2. Prove that " The sum of two odd numbers is even: using direct proof.

A:

Q: Determine the term involving y5 in the expansion of (3y +).

A: Please refer the attached image for complete solution. THANK YOU.

Q: Rationalize the denomi 20 5/10 - 7/5

A: 20510-75

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: 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: Use mathematical induction to prove that +1+2+4+...+2"-² = 2"- for all 2 -- positive integers . п.

A: We need to prove using mathematical induction given statement First we will show statement holds for…

Q: Prove the formula for the sum of the first positive integers:

A:

Q: Let ab5c be all four-digit multiples of number 36 with different digits. If someone found 7 numbers…

A: Let ab5c be all four digit multiples number 36 with different digits.   As 36= 4x9, so the number…

Q: Give a combinatorial interpretation of the coefficientof x4 in the expansion (1 + x + x2 + x3 +⋯)3.…

A:

Q: How many positive integers less than 1,000,000 have a digit sum of 5?

A: We have to find the number of positive integers 1,000,000 have a digit sum of 5. The number less…

Q: How many 5-digit even numbers can be formed using the digits 2, 4, 6, 7, 9 if no repetition is…

A: Given:

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: How do we proof that 2logn = n?

A:

Q: Prove through mathematical induction that...

A: Mathematical induction is a method of proving a statement, formula, or expression. In this, the…

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: What two nonnegative real numbers with a sum of 23 have thelargest possible product?

A: we have to find the two non negative real number with a sum of 23 which will have largest possible…

Q: Determine the smallest integer that can be expressed as a sum of two primes in three different ways

A: Let us have smallest prime numbers  are, 2,3,5,7,11,13 17,19.....

Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

A:

Q: how to prove √6 is irrational

A:

Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: The 4th term of an ap is zero. Prove that 25th term of the AP is three times its 11th term

A:

Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

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

A:

Q: 5. Give a recursive definition of the set of positive integers that are multiples of 5.

A: Solution

Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

A:

Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: prove its true by mathematical induction

A: The mathematical induction technique is used to prove the validity of the statement for every…

Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: Find the recursive and explicit rule for -1, 3, 7, 11,... Recursive rule

A: This is a problem of series . Based on concepts and formulas we will answer the question.

Q: How many positive integers not exceeding 3000 are divisible by exactly two of 4, 10, 12? (Steps…

A:

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

A:

Q: divisible by 3

A:

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: Find the average of the first 50 natural numbers is……………….

A:

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

A:

Q: Determine the smallest integer that can be expressed as a sum of two primes in three different ways.

A: The number is 22 and all others smallest integer can not be written as sum of two primes in three…

Q: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

A: Use Mathematical Induction to show that 52n -1is divisible by 8,   ∀ ∈ n N.

Q: Fermat's theorem is true for only prime number. إختر واحداً.

A:

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Find the sum of all  natural numbers between 250 and 1000 which are exactly divisible

A:

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: Fermat's theorem is not true for some prime number. اختر واحد

A: False