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# Basic Algebra

### Problems in Arithmetic and Geometrical Series

Three numbers are in arithmetic progression. The sum of the three numbers is 30 and the sum of their squares is 398. What are the three numbers? b) An arithmetic series is such that its first term is a and its third term is b. The sum of the first n terms is Sn . Find S4 in terms of a and b. Given that S4, S5 , S7 are cons

### Binomial Expansions

Find the following. (Questions held on attachment) Find the following: a) The coefficient of x in the expansion of b) The coeffiecient of x^3yz^4 in the expansion of c) The coefficient of x^3 in the expansion of d) The coefficient of x^5 in the expansion of is equal to the coefficient of x^4 in the

### Induction

Prove by induction where n is a positive integer. (The questions are attached).

### Solve the Quadratic Equation Step-by-Step

Solve this equation: 3x²- 5x -1 = 0.

### Find the common logarithm of

Please see the attached file for full problem description.

### Logarithm Value Deduction

Find the value of x: log_3 7.2 = x

### Dependent Variables Equations

Find a new dependent variable such that the equation becomes linear in that variable. Then solve the equation: 1/(y^2 + 1) y' + 2/x tan^-1 y =2/x

### Exact Equations Implicit Differentiation

First determine if the equation is exact. If it is exact, find the general solution, or at least a relation that defines the solutions implicitly: [cos(x^2 + y) - 3xy^2]y' + 2x cos(x^2 + y) - y^3=0

### Find the General Solution of the Equation

Solve for either the general solution or relation. xy' - y = x(1 + e^(-y/x)).

### Algebra problem and rectangles

If the perimeter of a rectangle is 10 inches, and one side is one inch longer than the other, how long are the sides? Can you show me the steps to take to work out this and similar problems.

### Investigation (Diagonals in a Rectangle)

Diagonals in a Rectangle. In the case of a 2 X 2 rectangle, or a 3 X 5 rectangle, we can simply count. However, can we make a decision about a 100 X 167 or a 3600 X 288 rectangle? In general, given an N X K rectangle, how many grid squares are crossed by its diagonal?

### Algebra - find the equation of a line

Find the equation of a line passing through (1,-3) and parallel to x+y=2.

### Equation of a Perpendicular Line

Find the equation of a line L which passes through the point (1,4) and is perpendicular to the line: 6x + 3y=12.

### Algebra - Circle center, radius, equation

The endpoints of the diameter of a circle are P=(-3,2) and Q=(5,-6) Find: (i) the center of the circle (ii) The radius of the circle (iii) the equation of the circle.

### Fibonacci Numbers for Mathematical Inductions

Application of Mathematical Induction It is an application of Mathematical Induction in proving the relations of Fibonacci Numbers. To prove: F 2n+1 - Fn Fn+2 = (-1) n for n > or, = 3.

### Logarithms and exponents

Logarithmic and Exponential Equations. See attached file for full problem description.

### Manipulated Logarithm Equations

Solve for x: log2x=8 Write as a single logarithm: 5lnx-1n(x+1)

### Logarithmic Equation: Solve for X

Log(x) means in this problem a log of base 3. ie, log(3)=1 sqrt(x) means the square root of x. ie, sqrt(25)=5 Solve the following equation for x: x^[log(9x)]=3sqrt(x).

### Partial Fraction Decomposition (using Quadratic Factors).

Solve the integral using partial fraction decomposition. This example has a denominator that is the product of quadratics. Example 1) S (x2 + x +1)/[(x2 + 3x +1)(x2 +4x +2)] dx

### Simplfy to a single rationale expression

2 - 13 divided by negative 76 + 15

### Simplify the Expression Divided

D divided by ab(c to the 4th) plus c divided by a(b to the third) c

### Mixing Problem : Exponenetial Functions

A 5000 gallon aquarium is maintained with a pumping system that circulates 100 gallons of water per minute through the tank. To treat a certain fish malady, a soluble antibiotic is introduced into the inflow system. Assume that the inflow concentration of medicine is 10te-t/50 oz/gal, where t is measured in minutes. The well-

### Simplifying Algebraic Expressions

I will use the ^ sign for the squared sign. Write the following algebraic expression in its simplest form: x^ + 2x + 3x^ + 2 + 4x + 7

### Introductory Algebra: Simplifying Algebraic Expressions

Simplify the following algebraic expression: 3X - (2 - X). Could you please explain the steps that I take when simplifying algebraic expressions such as this one? Thanks.

### Five Step Problem Solving Method for Algebraic Problems with example provided.

Can you provide me with the basic steps for solving problems in algebra? Please provide an example (e.g., a problem solving for the variable x) using these steps.

### How old is Karley?

Karley is twice as old as Lana. Three years from now, the sum of their ages will be 42. How old is Karley?

### One Variable Algebraic Problem

1. Dan's father is 45. He is 15 years older than twice Dan's age. How old is Dan?

### Well-Ordering Axiom - Strong Induction

Prove the well-ordering Axiom by strong induction.

### Proof by Mathematical Induction : Planes, Lines and Regions

Suppose that n straight lines in the plane are positioned so that no two are parallel an no three pass throught the same point. Show that they divide the plane into 1/2(n^2 + n + 2) distinct regions.

### Constant Map Functions

A mapping %:A->B is called a constant map if there exists b.(b not) belonging to B such that %(a) = b. for all a belonging to A. Show that a mapping %:A->B is constant if and only if %\$=% for all \$:A->A