Compute o, (100), 02 (750), Hints: the divisor-sum function Or(n). When k is any number and n is a positive integer, let ok (n) denote the sum of the k" powers of the positive divisors of n. For example: o (10) = 1° + 2° + 5° + 10° = 1+1+1+1 0, (10) = 11 + 21 + 51 + 101 0, (10) = 12 + 22 + 52 + 10² = 1+ 4 + 25 + 100 = 130 and o3 (1024). 4 1+2+5+ 10 18

Q: Let denote the n-th prime number. Show that Pn Pn n=1 diverges.

A: Divergence of the series

Q: (4) Determine the generating function for the sequence ho, h1,..., where hn = #((a1, a2, a3, a4) |…

A:

Q: Find the least integer n such that (2x^3+x^2logx) is O(x^n).

A: The given expression is 2x3+x2logx. Concept used: Let f and g from the set of integers or the set of…

Q: n! diverges because an > 0. for all n > 1. True False

A:

Q: Using the generating function to determine the number of ways  18 identical balloons can be given…

A: Given that 18 identical balloons can be given to 3 children such that each child receives at least 5…

Q: Write out the first six terms of the sequence defined by the recurrence relation with the given…

A:

Q: nl diverges because an > 0, for all n 2 1. ) True False

A: Given statement is

Q: snip

A: Given recurrence relation, xn=5xn-1-6xn-2                  ...(1) And x1=1 , x2=5

Q: Prove that the K4 is not consecutive.

A:

Q: Solve the following recursion relation T(n) = n* T(n-1) if n>1 = 1.…

A: Explanation of the answer is as follows

Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Use a combinatorial argument to prove that EC(n, k) = 2". k=0

A: The Multiplication Principle If one event can occur in m ways and a second event can occur in n ways…

Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: 3. Let fn be the number of strings of length n using the letters A, B, C, D, such that there are an…

A:

Q: Prove that then the sequence {"(1+i) belongs to l1. n3"

A:

Q: Calculate the coefficient of x7y5zx7y5z in (x+y+z)13(x+y+z)13. (b) In her discrete math class,…

A:

Q: Generate the first six terms of the sequence defined by the recurrence relations. an= nan-1-an-2,…

A: The recurrence relation is, an=nan-1-an-2 The first 2 terms are -1 and 0

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: Let a, be the number of strings of symbols {a, b, c, 1, 2, 3} of length n, such that exactly one of…

A: Given: Only one of the character is a numeric digit from the set(1,2,3,a,b,c).

Q: Write the first four terms of the sequence {an} defined by the recurrence relation below. an+1=3an -…

A:

Q: Find k such that, 2 det 4 li0 -8 -5] -1 -7 = 510 k 0 .

A: Given det2-8-54-1-710k0 =510 we have to find k

Q: Suppose that m and n are positive integers with m > n and f is a function from {1, 2,…, m} to {1,…

A: Consider the provided question, The function f is one to one if and only if f(a) = f(a) implies that…

Q: Ahmet borrows n TL from Burak. Starting from today, Ahmet pays back to Burak 2 TL or 3 TL at each…

A: Given that Ahmet borrows n TL from Burak. Starting from today, Ahmet pays back to Burak 2 TL or 3 TL…

Q: Find {2), f3), R4), and {5) if fis defined recursively by f0) = -1, A1) = 2, and {(+1) = Jtn}*s(n –…

A:

Q: 4. Find the product-of-sums expansions of the Boolean function F (x,y,z) that equals 1 if and only…

A: The complete solutions are given below

Q: Suppose that fis defined recursively by: f(0) = 5 and f(n+1)=2 fn+5. Find A(1), A(2), (3) and A(4)?…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Show that yn for all n=1, 2₁- ✓

A:

Q: Consider the sequence {ax} defined by a1 = 3, and ak+1 = ak + 10 for k 2 1. Compute a7.

A: Given  a1=3 and ak+1=ak+10 , k≥1to finda7

Q: Consider the sequence (am)neN defined by 1 A2m -1 for all m E N. A2m-1 m m ||

A: Given that (an)n∈N be the sequence defined asa2m-1=1ma2m=-1m

Q: Find a generating function for the sequence (ao, a1,...) where a, is the number of nonnegative…

A: To find- Find a generating function for the sequence a0, a1, ... where ar is the number of…

Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: Consider the following recursively defined sequence. fk = fk-1+ 24, for each integer k 2 2 f, = 1…

A:

Q: 4. Show that P(n,k)= P(n – 1, k)+ k · P(n – 1,k – 1). Using Combinatorial Proof.

A:

Q: Consider the recurrence relation:   T(n) = { a, if n == 1 { b + cN + T(n-1), if N >= 2…

A: The closed form of recurrence relation is given by a general formula containing nth term or n-1 th…

Q: Write out the first six terms of the sequence defined by the recurrence relation with the given…

A: Given:                     y0=0y1=1yn=yn-1-yn-2 for n≥2   To determine:                       The…

Q: Find the ordinary generating function for the sequence 0,0,0,0,1,1,1...(Hint: it is not a…

A: We need to find an ordinary generating function. The sequence is 0,0,0,0,1,1,1,.... Consider:…

Q: n! diverges because an > 0. for alln2 1. True False

A:

Q: 15)- Estimate f' (4),and f"(4) using central difference formulas of order o(h2) for the data (0,1),…

A:

Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

A:

Q: Use generating functions to determine the number of 9-digit sequences of the digits 1,2,3,4,5,and 6…

A: We need to find ,  number of 9-digit sequences of the digits 1,2,3,4,5,and 6 in which each even…

Q: Let fn be the number of strings of length n using the letters A, B, C, D, such that there are an…

A:

Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: (4) Determine the generating function for the sequence ho, h1, ..., where hn = #((a1, a2, a3, a4) |…

A: By definition of Generating Function,                The Generating function for the sequence…

Q: show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

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 positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.