### Be Reasonable

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

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?

### 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.

# Speedy Summations

##### Stage: 5 Challenge Level:

In the video below, Alison works out $\sum_{i=1}^{10} i$.

How could you adapt her method to work out the following sums?

• $\sum_{i=1}^{100} i$

• $2+4+6+\dots+96+98+100$

• $\sum_{k=1}^{20} (4k+12)$

• $37+42+47+52+\dots+102+107+112$

• The sum of the first $n$ terms of the sequence $a, (a+d), (a + 2d), (a + 3d) \dots$

After how many terms would $17+21+25+\dots$ be greater than $1000$?

Can you find the sum of all the integers less than $1000$ which are not divisible by $2$ or $3$?

Can you find a set of consecutive positive integers whose sum is 32?