Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the integers:    1^5+2^5+...n^5=1/6 n^6+1/2 n^5 +5/12 n^4 - 1/12 n^2

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: 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: How many terms are there in the expansion of (a+b)48 after like terms are collected?

A: This question is related to binomial expansion. We will solve it using given information.

Q: Without using any theorems from the book, prove the following statement: The sum of any two odd…

A:

Q: Suppose the repeating in 0.020020200202002.... continues forever. Is this number rational?

A: We can not write the number 0.020020200202002... in the form  p/q where p is an integer and q is a…

Q: 21 j=3

A: Use the formula for the sum of n natural numbers.

Q: What is the exponent on the x variable of the 3st/nd/rd/th term in the expansion of 17 (x2 +3y)" ?…

A: using formula:-

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

A:

Q: Show all formula, substitution and calculations. Round off to 2 decimals. In how many ways can…

A: The collection of items or elements arranged in a different way with ordering is called Permutations…

Q: What is the summation from n = 1 to infinity, of  (3n+1) / (5n + 4) ?

A:

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: Use the rules for sums of powers of integers to compute the sum. 17

A: Given : the summation of k2, k from 3 to 17.

Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

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 number whose decimal reoresentation repeats is a rational number. true or false

A: A number is rational if and only if its decimal representation is repeating or terminating (i.e.…

Q: Find the sum of the positive even integers up to and including 50.

A:

Q: Use mathematical induction to prove that any amount of postage that is an integer amount greater…

A: We have to prove that any amount of postage that is an integer amount greater than 53 cents can be…

Q: Determine the nth term formula for the number of square tiles in the nth figure. an %3D as a2 az

A: Number of tiles in a1 is 2, n=1 Number of tiles in a2 is 4, n=2 Number of tiles in a3 is 6, n=3…

Q: (nt) n=1

A: we have to find the sum

Q: 1. Find the sum of the first 50 even positive integers.

A:

Q: prove that "The Sum of two odd numbers is even: using direct. Note:- 1 want the ansawer on a paper.

A: The sum of two odd numbers is even. Proof :          Properties odd and even integer.          If…

Q: Find the sum of all integers that are multiples of 3 from 4 to 100.

A: We have to find the sum of all integers that are multiples of 3 from 4 to 100 Then, If A denotes the…

Q: Use the rules for sums of powers of integers to compute the sum. 18 Σ; j=5

A: We can find the answer in the following steps.

Q: Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

A:

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: Find the sum of all positive integers, from 5 to 1555 inclusive, that are divisible by 5.

A:

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: Find the sum of all even integers from 10 to 500

A: Sum of all even integers from 10 to 500 .

Q: Which numbers in Z17 are relatively prime to 17? Enter your answer as a comma separated list of…

A: The objective of the question is determine all al relatively prime to 17 in Z17 .

Q: Use Dirichlet's theorem to show that there are infinitely many primes whose dicimal expansion ends…

A:

Q: 3. Show that 3.141592592592592... is a rational number, by expressing it as a ratio of two integers.

A:

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

A: If a number n is not divisible by 2 then it is an odd number.  If a number n divisible by 2 then it…

Q: The number of primes of the form n³ – 1 is 1) 1 2) 2 3) Infinite

A: To choose correct option

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

A:

Q: ition to prove the n positive number. of Your Your answe that 1/x 00 as

A:

Q: 7, Evaluate the integral Express numbers in exact form.use symbolic notation and fractions where…

A:

Q: A DIGITAL CHIP MADE BY YOUR. MICROELECTRONICS DIVISION, COUNTS CONTINUOUSLY IN THE SEQUENCE: 1024…

A: The counts of the chip are 1024,2048,4096,8192,.... Now, we can see that 20481024=40962048=2 Each…

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

A:

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: Use the rules for sums of powers of integers to compute the sum. 21 Σ; j=5

A:

Q: Find the sum of all natural numbers n such that 2021-n/99 is a natural number ?

A:

Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: Find mZD. 6. D Write your answer as an integer or as a decima mZD = %3D

A: We have a right-angled triangle right angled at F we have to find angle D

Q: Find the sum of the first 19 odd positive integers. Need Help? Read It

A:

Q: For n > 3, prove that the numbers ±5, ±52, · reduced residue system modulo 2". form a

A: A subset R of the integers is called reduced a residue system modulo n if (i) gcdr,n =1 for each r…

Q: Find the sum and product of each of these pairs of numbers (20CBA)16 , (A01)16- Express your answers…

A:

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

A: Binomial theorem, (a-b)n=C0nan-C1nan-1b+C2nan-2b2-C3nan-3b3+...+Cn-1na1bn-1+Cnnbn To find binomial…

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

A: False