Creating and manipulating expressions and formulae

After performing some operations, what number is your answer always a multiple of?

Can you find the value of t in these equations?

Can you find a rule which connects consecutive triangular numbers?

Can you work out the value of x in this 'power-full' equation?

Each letter stands for a different digit, and S is non-zero. Which letter has the lowest value?

Derive an equation which describes satellite dynamics.

What does the sum of these three numbers tell us about their product?

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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.

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. Prove that all terms of the sequence are divisible by 6.