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. . . .

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?

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

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.

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

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

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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

Relate these algebraic expressions to geometrical diagrams.

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

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.

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?

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?

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

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

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

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.

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

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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.

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

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

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

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}.

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.

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

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?

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.

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

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

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

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.

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.

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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

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.

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Kyle and his teacher disagree about his test score - who is right?

Five equations... five unknowns... can you solve the system?

Can you explain why a sequence of operations always gives you perfect squares?

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

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?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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. . . .

A task which depends on members of the group noticing the needs of others and responding.

What is the total number of squares that can be made on a 5 by 5 geoboard?