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

Fibonacci Factors

Age 16 to 18
Challenge Level

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

How Many Solutions?

Age 16 to 18
Challenge Level

Find all the solutions to the this equation.

Absurdity Again

Age 16 to 18
Challenge Level

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Particularly General

Age 16 to 18
Challenge Level

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

Poly Fibs

Age 16 to 18
Challenge 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.

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

More Polynomial Equations

Age 16 to 18
Challenge 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.

Always Perfect

Age 14 to 18
Challenge Level

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

Always Two

Age 14 to 18
Challenge 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.

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?

Perfectly Square

Age 14 to 16
Challenge Level

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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?

Enriching Experience

Age 14 to 16
Challenge Level

Find the five distinct digits N, R, I, C and H in the following nomogram

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.

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.

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.

Multiplication Square

Age 14 to 16
Challenge Level

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

Algebra from Geometry

Age 11 to 16
Challenge Level

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

Reciprocals

Age 16 to 18
Challenge Level

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

Puzzling Place Value

Age 14 to 16
Challenge Level

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

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.

Consecutive Squares

Age 14 to 16
Challenge Level

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Number Rules - OK

Age 14 to 16
Challenge Level

Can you produce convincing arguments that a selection of statements about numbers are true?

Really Mr. Bond

Age 14 to 16
Challenge 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 16
Challenge Level

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

Difference of Two Squares

Age 14 to 16
Challenge Level

What is special about the difference between squares of numbers adjacent to multiples of three?

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.

DOTS Division

Age 14 to 16
Challenge 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}.

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?

Triangles Within Triangles

Age 14 to 16
Challenge Level

Can you find a rule which connects consecutive triangular numbers?

Triangles Within Squares

Age 14 to 16
Challenge Level

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

Incircles

Age 16 to 18
Challenge Level

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Interactive Number Patterns

Age 14 to 16
Challenge Level

How good are you at finding the formula for a number pattern ?

Magic Sums and Products

Age 11 to 16

How to build your own magic squares.

Square Number Surprises

Age 14 to 16
Challenge Level

There are unexpected discoveries to be made about square numbers...

Triangles Within Pentagons

Age 14 to 16
Challenge Level

Show that all pentagonal numbers are one third of a triangular number.

Pythagoras Perimeters

Age 14 to 16
Challenge Level

If you know the perimeter of a right angled triangle, what can you say about the area?

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?

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.

Archimedes and Numerical Roots

Age 14 to 16
Challenge Level

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

What's Possible?

Age 14 to 16
Challenge Level

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

Series Sums

Age 14 to 16
Challenge Level

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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. . . .

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.

Pareq Calc

Age 14 to 16
Challenge Level

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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. . . .

Sums of Squares

Age 16 to 18
Challenge Level

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

Unusual Long Division - Square Roots Before Calculators

Age 14 to 16
Challenge Level

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

Look Before You Leap

Age 16 to 18
Challenge Level

Relate these algebraic expressions to geometrical diagrams.