Other tags that relate to Binomial
Creating and manipulating expressions and formulae. Mathematical reasoning & proof. Polynomial functions and their roots. Working systematically. Identities. Making and proving conjectures. Generalising. Binomial Theorem. Indices. Inequalities.There are 172 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
Binomial
Age 16 to 18 Challenge Level:
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
Unit Interval
Age 14 to 18 Challenge Level:
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Particularly General
Age 16 to 18 Challenge Level:
By proving these particular identities, prove the existence of general cases.
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Quadratic Harmony
Age 16 to 18 Challenge Level:
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
Diverging
Age 16 to 18 Challenge Level:
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
Sixational
Age 14 to 18 Challenge Level:
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
Interpolating Polynomials
Age 16 to 18 Challenge Level:
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
Mechanical Integration
Age 16 to 18 Challenge Level:
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Pair Squares
Age 16 to 18 Challenge Level:
The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
Telescoping Functions
Age 16 to 18
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
Look Before You Leap
Age 16 to 18 Challenge Level:
Relate these algebraic expressions to geometrical diagrams.
Three Ways
Age 16 to 18 Challenge Level:
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
And So on - and on -and On
Age 16 to 18 Challenge Level:
Can you find the value of this function involving algebraic fractions for x=2000?
Polynomial Relations
Age 16 to 18 Challenge Level:
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
Rational Roots
Age 16 to 18 Challenge Level:
Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.
Transitivity
Age 16 to 18
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
Sums of Squares and Sums of Cubes
Age 16 to 18
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.
Integral Inequality
Age 16 to 18 Challenge Level:
An inequality involving integrals of squares of functions.
Little and Large
Age 16 to 18 Challenge Level:
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
Sperner's Lemma
Age 16 to 18
An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.
An Alphanumeric
Age 16 to 18
Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.
To Prove or Not to Prove
Age 14 to 18
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Exhaustion
Age 16 to 18 Challenge Level:
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Proof Sorter - Quadratic Equation
Age 14 to 18 Challenge Level:
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
Can it Be?
Age 16 to 18 Challenge Level:
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Dalmatians
Age 14 to 18 Challenge Level:
Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.
Plus or Minus
Age 16 to 18 Challenge Level:
Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
Janine's Conjecture
Age 14 to 16 Challenge Level:
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
Leonardo's Problem
Age 14 to 18 Challenge Level:
A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
Yih or Luk Tsut K'i or Three Men's Morris
Age 11 to 18 Challenge Level:
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
Euclid's Algorithm II
Age 16 to 18
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
Iffy Logic
Age 14 to 18 Challenge Level:
Can you rearrange the cards to make a series of correct mathematical statements?
A Long Time at the Till
Age 14 to 18 Challenge Level:
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Water Pistols
Age 16 to 18 Challenge Level:
With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?
Golden Eggs
Age 16 to 18 Challenge Level:
Find a connection between the shape of a special ellipse and an infinite string of nested square roots.
Target Six
Age 16 to 18 Challenge Level:
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Without Calculus
Age 16 to 18 Challenge Level:
Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.
Some Circuits in Graph or Network Theory
Age 14 to 18
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Basic Rhythms
Age 16 to 18 Challenge Level:
Explore a number pattern which has the same symmetries in different bases.
Tetra Inequalities
Age 16 to 18 Challenge Level:
Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.
Tree Graphs
Age 16 to 18 Challenge Level:
A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .
Polite Numbers
Age 16 to 18 Challenge Level:
A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?
On the Importance of Pedantry
Age 16 to 18
A introduction to how patterns can be deceiving, and what is and is not a proof.
An Introduction to Number Theory
Age 16 to 18
An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)
NRICH is part of the family of activities in the Millennium Mathematics Project.