# Advanced Problem Solving Module 23

### Advanced Problem Solving Module 23

Summing series comes up in many STEP and other advanced mathematics examinations questions, so it's important to be familiar with the key concepts. This module offers some reminders and practice.

Sequences are everywhere in mathematics! In these problems you can explore some sequences, learn how to represent them, and how to calculate sums of series.

##### Age 16 to 18 Challenge Level:

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.

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

Is it true that a large integer m can be taken such that: 1 + 1/2 +
1/3 + ... +1/m > 100 ?