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Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

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In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?

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Summats Clear

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

Polite Numbers

Age 16 to 18 Challenge Level:

A polite number is a number which can be written as the sum of two or more consecutive positive integers. Find the two consecutive sums which produce the polite numbers $544$ and $424$.

How would you represent these sums using a number line? Use this visualisation approach to decide which consecutive sums would give rise to the polite numbers $1000$ and $1001$. Do these numbers arise as more than one consecutive sum? How do these numbers relate to the formula for the sum of an arithmetical progression?

Can you find any numbers which are not polite?

There is actually a rather simple rule which determines whether a given number is polite. Can you find this rule? Can you prove that this is the case?