Proof By Induction

Prove by induction (or any other method): For any natural n the following statement is true: (2n+1)+(2n+3)+(2n+5)+.....(4n-1)=3n^2

Word Problem - How many Cookies?

Dad made cookies. Dad ate 1 cookie. Dave ate 1/2 dozen cookies. Kate ate 1/2 of what Dave ate. Henry & Julie each ate 1/3 of what was left. Then Jake ate 1/2 of what was left. Mom ate 1 cookie. There was 1 cookie left. How many cookies were baked?

Functions: Variable Replacement Problem

The problem reads: If a @ b = 3a - b^2 find 2 @ (3 @ 1)

Summation Equality: Prove by Induction

I would be grateful if someone could show me how to prove by induction the attached formula.

The Maximum of the Function

Find the maximum of f(x,y)=xy such that x+3y=12

Algebra: Word Problem - Relative Velocity

A mo-ped can travel 60 miles in 2 hours less time than a bicycle can travel 50 miles. The mo-ped is traveling at a rate of 10 miles per hour faster than the bicycle. 1. How fast in mph is each traveling? 2. How long will it take each to travel their respective distance?

Inequality: Proof using Unit Vectors

Prove that (a1b1+a2b2+...+anbn)^2<=(a1^2+a2^2+...+an^2)(b1^2+b2^2+...+bn^2) where ai, bi are real numbers, i=1,2,...,n

Linear Algebra : Zero Matrix Proof

Let A be a 2 x 2 matrix with A^3 = O. Prove that A^2 = O. Where O is the zero matrix. (Note: I've said nothing about the invertibility of A, it may or may not be invertible).

Algebraic Equations : Reciprocal Equations

As provided by E. Galua theory the general algebraic equations for a polynomial of fourth order ax^4 + bx^3 + cx^2 + dx + f=0 (*) is the maximum order type of algebraic equations the solution to which one can write down in radical expressions. Among all the equations of fourth ...continues

Equalities : Summations

Clarification on equalities. Please see attached file.