Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, radicals, etc.

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3
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1answer
49 views

The unexpected, $\ 2^n$ is not of the form $2xy + x$?

When testing polynomials for items of an integer sequence, I find that $2^n$ is not of the form $2xy + x,\ n,\ x,\ y \in \mathbb N.\ $How to prove this?
1
vote
2answers
74 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
1
vote
1answer
102 views

is the decimal notation the “right” notation for arithmetic?

I am considering here the pre-decimal notations such as Roman numerals, Egyptian numerals etc. It seems reasonable that these must all be equivalent. And it seems that decimal notation (i.e. ...
0
votes
3answers
89 views

How to derive this inequality?

How to derive the following inequality for all positive integers $n \geq 2$? $$ \frac{n!}{n^n} \leq \left(\frac{1}{2}\right)^k,$$ where $k$ denotes the greatest integer less than or equal to $\dfrac ...
0
votes
2answers
71 views

How to derive these inequalities?

I can derive the inequalities $$ n^p < \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^p $$ for any positive integers $p$ and $n$. These actually follow from the identity $$b^p - a^p = (b-a)(b^{p-1} + ...
4
votes
8answers
449 views

$\left(\sqrt{8}+\sqrt{2}\right)^2$ = 18 why??

I would like to now why $$\left(\sqrt{8}+\sqrt{2}\right)^2$$ is equal to $18$? Please provide me the proccess. Thank you.
4
votes
3answers
1k views

Daily exercises to speed up my mental calculations?

When I was a kid in school my father prevented me from using a calculator when solving my math homeworks. However at that time I was not convinced as of why not to use such a useful tool! So I kept on ...
2
votes
2answers
24 views

How to obtain this upper bound on the summation from this inequality?

I can show that $$ \frac{1}{\sqrt{n}} < 2 (\sqrt{n} - \sqrt{n-1} ) $$ for $n \geq 1$. Now from this how to derive the following inequality? $$ \sum_{n=1}^m \frac{1}{\sqrt{n}} < 2\sqrt{m} - 1 $$ ...
7
votes
1answer
96 views

$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $

Show that $$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $$ Indeed, First let's show $7\mid x \text{ and } 7\mid y \Longrightarrow 7\mid x^2+y^2 $ we've $7\mid x ...
3
votes
1answer
174 views

Prove that $2+2=4$.

Before you might chastise this quesion, I understand that we all know $2+2=4$. But a while ago I just stumbled across this paper which formally proves that $2+2=4$: ...
1
vote
5answers
89 views

$\forall \ n \in \mathbb{N}: 6\mid 5n^3+n$

Show $\forall \ n \in \mathbb{N}: 6\mid 5n^3+n$ Indeed, we've to show $5n^3+n=0\pmod 6 $ note that for $n\in \{0,1,2,3,4,5\}$ we've: $n^{3}=n\pmod 6$ then $5n^{3}=5n\pmod 6 \implies ...
-1
votes
1answer
38 views

Really Simple Re Weighting Question

I am having a major brain cramp and cannot remember how to to this. In a perfect world I have 7 types of activities that I could have participated in. These 7 activities each have a weight that ...
1
vote
5answers
65 views

$n \in \mathbb{N} \ 5|\ 2^{2n+1}+3^{2n+1}$

show for all $n \in \mathbb{N}$, $$5|\ 2^{2n+1}+3^{2n+1}$$ Indeed, we've to show that : $2^{2n+1}+3^{2n+1}=0[5] $ note that $2^{2n+1}+3^{2n+1}=2.4^n+3.9^n= $
6
votes
2answers
132 views

Solve equation with $x$ in the exponents

How can I solve $$8(9^x)+3(6^x)-81(4^x)=0 $$ for $x$ using elementary methods. Thanks a lot!
0
votes
2answers
118 views

A man invests an amount of $ ₹ 15860 $ in the names of his three sons A, B & C in such a way that they get the same interest..

A man invests an amount of $ ₹ 15860 $ in the names of his three sons A, B & C in such a way that they get the same interest after $2, 3$ & $4$ years respectively. If the rate of simple ...
14
votes
5answers
2k views

Is $ 5 $ nearer to $ 0 $ or $ 10 $?

My 6-year-old’s homework was “to find the nearest $ 10 $.” For example, $$ 42 \to 40 \quad \text{and} \quad 28 \to 30. $$ For $ 55 $, she answered “$ 50 $” and was marked wrong. How is this wrong? ...
0
votes
1answer
19 views

Help with a summatory.

Given $B=\{e_1,\dots,e_n\}$ a basis of a K-vector space $V$, and $f$ the only endomorphism such that $f(e_i)=\begin{cases} e_1-e_2, & \mbox{if } i=1 \\ -e_{i-1}+e_i-e_{i+1}, & \mbox{if } 2 ...
2
votes
1answer
35 views

Doubt about the domain of functions with odd roots.

Given the function: $f(x) = \sqrt[3]{x} \cdot \sqrt[5]{x-1}$, Wolfram claims the domain is $x\ge1$, the same of the $\sqrt[5]{x-1}$ part. My question is: why isn't the domain $\mathbb{R}$? I've ...
1
vote
2answers
53 views

Question on Clocks

Three watches are set together. The first gains $5$ minutes a week, the second gains $8$ minutes a week, whilst the third loses $4$ minutes a week. When will they again indicate the same time? ...
2
votes
3answers
110 views

How to solve $\left(\dfrac{5}{3}\right)^3\left(-\dfrac{3}{5}\right)^2$ [closed]

I need help in solving this problem (sorry I didn't know how to write it on here).
1
vote
5answers
55 views

Simple Mathematics equation

My friend shared me an mathematics equation which is following $$3+3\cdot3-3\cdot0=?$$ What is the answer of this equation?
0
votes
2answers
80 views

Cartesian coordinates of lemniscate

A problem from Spivak's Calculus: Sketch the graph of the lemniscate $$r^2 = 2a^2\cos 2 \theta.$$ Find an equation for its cartesian coordinates. Show that it is the collection of ...
6
votes
5answers
434 views

Simple solving Skanavi book exercise: $\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}$

Simple way to solve this exercise $$ x = \sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}} $$
2
votes
3answers
1k views

HARD Arithmetic Progression Problem

The first term is 8 and the common difference is d, where d doesn't = 0. The first term, the fifth term, and the eighth term of the progression are the first term, the second term and the third term, ...
1
vote
1answer
130 views

How to define multiplication in addition terms in monadic second order logic?

How to define multiplication in addition terms in monadic second order logic? meaning, having natural numbers variables, N sub-groups variables, successor function, negations, "for every", "there ...
0
votes
1answer
715 views

How to calculate arithmetic mean of log values

I am working with really small values of probabilities and that is why their log values are used. So for example, let probA and probB be some normal values of probabilities of two events and because ...
0
votes
1answer
44 views

$AM+GM+HM+RMS=66$

For two reals $a,b\in\mathbb{R}$ Arithmetic mean(AM), Geometric Mean(GM), Harmonic Mean(HM) and Root mean square (RMS) all are integers and $AM+GM+HM+RMS=66$ Find all such $a,b$ I have assumed ...
1
vote
3answers
66 views

Given a rational, how to find the integers whose quotient it is?

I haven't found an answer to this anywhere. Excluding brute force, given a rational $q$ in its decimal form ($1.47$, for example), is there a good algorithm to find integers $m$ and $n$ such that ...
1
vote
1answer
35 views

how many questions did the answer in each round?

yolanda went on a quiz show. the question in the first round were worth 6 points. the questions in second round were worth 10 points. yolanda answered a total of 5 questions and earned 34 points. how ...
2
votes
2answers
99 views

Calculate f(n+1)-f(n-1) based on f(n)???

Being: $$f(n) = \left(\frac{5+3\sqrt5}{10}\right)\cdot\left(\frac{1+\sqrt5}{2}\right)^n+\left(\frac{5-3\sqrt5}{10}\right)\cdot\left(\frac{1-\sqrt5}{2}\right)^n$$ Calculate: $$f(n+1)-f(n-1)\\ ...
7
votes
7answers
1k views

Is it possible to calculate the arithmetic mean from the geometric mean?

An estimation method would be acceptable, doesn't need to be exact (but obviously that would be preferable). I have a dataset of geometric means, need to calculate the arithmetic mean.
2
votes
3answers
41 views

I need help with an average question

My brother and I are going on vacation. We need to rent a car. We will rent the car for $10$ days. However, I will only be on the vacation for $5$ days. My brother will be on vacation for the full ...
2
votes
1answer
67 views

maths question for level 2 maths essential skills.

In Seville, Sue sees some wall tiles that she would like for her kitchen. The big tile measures $20 \text{ cm}\times 20 \text{ cm}$. The small tile measures $10 \text{ cm}\times 10 \text{ cm}$. How ...
0
votes
2answers
283 views

Prove that if $n^2 - 1$ is divisible by a prime number $p$ such that $n - 1$ is not divisible by $p$, then $n + 1$ is also divisible by $p$.

If this proposition is false, please give at least $3$ counter-examples, and try to modify the proposition so that it becomes true. If the proposition is true, please try to prove this even more ...
9
votes
4answers
472 views

Calculate the product

$$\left( 1 + \frac{1}{2} \right) \left( 1 + \frac{1}{3} \right) \left( 1 + \frac{1}{4} \right) ... \left( 1 + \frac{1}{99} \right) $$ Of course I could do it factor for factor but there has got to be ...
0
votes
1answer
17 views

Show that if $a=bq+r $ and $d|a$ and $d |b $, then $d|a-bq $

Show that if $a=bq+r $ and $d|a$ and $d |b $, then $d|a-bq $ That is show that if $d $ divides $a $ and $a=bq+r $ then $d $ divides $a-bq $. Here $d |a $ means "d divides a", that is $ a=dk $ where ...
3
votes
4answers
1k views

I want to learn math from ground-up, basic to advanced, beginner to expert

I want to learn math. I've learned math long time ago, but i hardly remember anything. I really want to relearn but have no idea where to begin. I want to learn math by reading through good books, ...
0
votes
7answers
199 views

How to derive $\;a^3 + b^3 = (a + b)(a^2 - ab + b^2)\;?$

I'm studying in 8th grade in India. I saw the question in my FIITJEE textbook, but it was not shown how to derive it. Please help. My mid phase examinations are coming up.
1
vote
1answer
45 views

How to express the relationship between two numbers by dividing?

I recently edited an image to bring it from 194px high to 140px high I'd like to describe how much shorter the image is (or, how different the two are in general) There are two ways to compare the ...
3
votes
0answers
35 views

question in product

can any expert just check my solution You bought a car for $\$2500$ down and made payments of $\$299.50$ each month for $36$ months. (a) Find the amount of the payments over the $36$ months. (b) Find ...
10
votes
2answers
512 views

Multiplying using reciprocal, addition and subtraction

Let $a,b\in\mathbb{R}$. How can we compute $a\times b$ using only the following operations only (with any reals) : $\frac{1}{*}$ (inverse) $*+*$ (addition) $*-*$ (subtraction) ?
0
votes
0answers
16 views

How can I know the number of steps required to get to a single digit number when adding the digit of a number repeatedly?

For example, 82934 would give 2 because 82934 -> (8+2+9+3+4)=26 -> (2+6)=8 I want to know the number of steps depending on the number. If it's not possible, I would like an approximation of it. If ...
4
votes
3answers
1k views

Can you give me a visual representation on 1/6 of 4/5 to get 4/30?

I'm trying to understand fractions more by using a visual representation, and I having a hard time making a visual representation of the 1/6 of 4/5 and getting a visual understandable representation ...
3
votes
1answer
127 views

Alphametics Question

In the figure below, each distinct letter represents a unique digit such that the arithmetic sum holds. If the letter L represents 9, what is the digit represented by the letter T? ...
0
votes
1answer
75 views

Why is the word “of” equivalent to multiplying in fraction word problems?

I know this is a very easy problem, but I'm having a hard time getting my head around this concept, consider this example from a book. *Jerry bought a pie and ate 1⁄5 of it. Then his wife Doreen ate ...
0
votes
2answers
40 views

How to calculate value of x in (1875+y):(x+y)=165

I need to solve this equation but every time I get a different result.Ive tried "find out $x$" calculators online but they fail with "error" or "invalid [...]". The equation is $(1875+y):(x+y)=165$ ...
0
votes
1answer
25 views

An implication between two statements concerning integers

Does "Every non-empty bounded below set of integers has a smallest element" $\implies$ "If $m,n$ are integers with $m>n$ , then $m-n\ge1$" ? If not then what additional assumption is needed ...
1
vote
0answers
42 views

Terminology: Name for general, non-differential equations over $\mathbb{R}^n$

Let $x \in \mathbb{R}^n$ be a variable, and $f_1, \ldots , f_n : \mathbb{R}^n \mapsto \mathbb{R}$ be a sequence of functions, each having a closed form expression. Assume that we want to solve the ...
-1
votes
1answer
23 views

How to add and subtract functions? [closed]

Given: $(f+g)(x)= 4x+5$ and $(f-g)(x)= -6x+3$ find $f(x)$ and $g(x)$ algebraically. Please help!
2
votes
1answer
85 views

How to solve 2 ÷ 2 ÷ 2 ? ${}{}{}{}$

$$2 ÷ 2 ÷ 2 = (2 ÷ 2) ÷ 2 \ \ \text{OR}\ \ 2 ÷ (2 ÷ 2) ?$$ Is there any standard rule which is world wide accepted for solving this type of expressions? If I process the expression from left to ...