## Unit 05: Miscellaneous Series (Solutions)

This is a fifth unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions.

After reading this unit the students will be able to

- Recognize sigma notation.
- Find the sum of
- The first $n$ natural numbers.

- The squares of first $n$ natural numbers.
- The cubes of first $n$ natural numbers.
- Define arithmetico-geometric series.
- Find sum to n terms of the arithmetico-geometric series.
- Define method of differences.
- Use this method to find the sum of $n$ terms of the series whose differences of the consecutive terms are either in arithmetic or in geometric sequence.
- Use partial fractions to find the sum to $n$ terms and to infinity the series of the type $\dfrac{1}{a(a+d)}+\dfrac{1}{(a+d)(a+2 d)}+ \cdots $