# Resources tagged with: Creating and manipulating expressions and formulae

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Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae ### Diverging

##### Age 16 to 18Challenge 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. ### Unit Interval

##### Age 14 to 18Challenge Level

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product? ### Absurdity Again

##### Age 16 to 18Challenge Level

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b? ##### Age 16 to 18Challenge 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. ### Pair Squares

##### Age 16 to 18Challenge 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. ### Always Two

##### Age 14 to 18Challenge Level

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2. ### Reciprocals

##### Age 16 to 18Challenge Level

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2. ### 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. ### Mechanical Integration

##### Age 16 to 18Challenge Level

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values. ### Always Perfect

##### Age 14 to 18Challenge Level

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. ### Interpolating Polynomials

##### Age 16 to 18Challenge 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. ### Binomial

##### Age 16 to 18Challenge Level

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn ### And So on - and on -and On

##### Age 16 to 18Challenge Level

Can you find the value of this function involving algebraic fractions for x=2000? ### Fibonacci Factors

##### Age 16 to 18Challenge Level

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3? ### Look Before You Leap

##### Age 16 to 18Challenge Level

Relate these algebraic expressions to geometrical diagrams. ### Simplifying Doughnut

##### Age 14 to 18Challenge Level

An algebra task which depends on members of the group noticing the needs of others and responding. ### Three Ways

##### Age 16 to 18Challenge Level

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra. ### Poly Fibs

##### Age 16 to 18Challenge Level

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys. ### Polynomial Interpolation

##### Age 16 to 18Challenge Level

Can you fit polynomials through these points? ### Polynomial Relations

##### Age 16 to 18Challenge 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. ### System Speak

##### Age 16 to 18Challenge Level

Five equations... five unknowns... can you solve the system? ### Particularly General

##### Age 16 to 18Challenge Level

By proving these particular identities, prove the existence of general cases. ### Sixational

##### Age 14 to 18Challenge 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. . . . ### How Many Solutions?

##### Age 16 to 18Challenge Level

Find all the solutions to the this equation. ### Leonardo's Problem

##### Age 14 to 18Challenge 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? ### Complex Partial Fractions

##### Age 16 to 18Challenge Level

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers. ### More Polynomial Equations

##### Age 16 to 18Challenge Level

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively. ### Little and Large

##### Age 16 to 18Challenge 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? ### Janine's Conjecture

##### Age 14 to 16Challenge 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. . . . ### ' Tis Whole

##### Age 14 to 18Challenge Level

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed? ### Sums of Squares

##### Age 16 to 18Challenge Level

Can you prove that twice the sum of two squares always gives the sum of two squares? ### Enriching Experience

##### Age 14 to 16Challenge Level

Find the five distinct digits N, R, I, C and H in the following nomogram ### Magic Squares for Special Occasions

##### Age 11 to 16

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line. ### W Mates

##### Age 16 to 18Challenge Level

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T. ### Triangles Within Squares

##### Age 14 to 16Challenge Level

Can you find a rule which relates triangular numbers to square numbers? ### DOTS Division

##### Age 14 to 16Challenge Level

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}. ### Perfectly Square

##### Age 14 to 16Challenge Level

The sums of the squares of three related numbers is also a perfect square - can you explain why? ### Really Mr. Bond

##### Age 14 to 16Challenge Level

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise? ### Pair Products

##### Age 14 to 16Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? ### Triangles Within Triangles

##### Age 14 to 16Challenge Level

Can you find a rule which connects consecutive triangular numbers? ### Multiplication Square

##### Age 14 to 16Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? ### Triangles Within Pentagons

##### Age 14 to 16Challenge Level

Show that all pentagonal numbers are one third of a triangular number. ### What's Possible?

##### Age 14 to 16Challenge Level

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make? ### Matchless

##### Age 14 to 16Challenge Level

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ? ### Magic Sums and Products

##### Age 11 to 16

How to build your own magic squares. ### Algebra from Geometry

##### Age 11 to 16Challenge Level

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares. ### Puzzling Place Value

##### Age 14 to 16Challenge Level

Can you explain what is going on in these puzzling number tricks? ### Plum Tree

##### Age 14 to 18Challenge Level

Label this plum tree graph to make it totally magic! ### Harmonic Triangle

##### Age 14 to 16Challenge Level

Can you see how to build a harmonic triangle? Can you work out the next two rows? ### Marbles in a Box

##### Age 11 to 16Challenge Level

How many winning lines can you make in a three-dimensional version of noughts and crosses?