Chapter 8: Sequences and Series

A sequence is an ordered list of numbers; a series is what you get when you add them up. These two ideas are everywhere in mathematics , from the interest on a savings account to the partial sums that approximate $\pi$, from population models to the digital representation of audio. In Class XI we focus on the two simplest and most useful families: arithmetic and geometric.

An arithmetic progression (AP) has a constant difference between consecutive terms. A geometric progression (GP) has a constant ratio. These two patterns govern almost every real-world growth process. Linear growth (filling a tank at a steady rate) is arithmetic. Exponential growth (compound interest, bacterial cultures) is geometric. Mastering them is mastering the algebra of change.

The chapter develops three sets of tools. First, formulas for the $n$-th term and the sum of the first $n$ terms. Second, means: the arithmetic mean (AM), the geometric mean (GM), and the celebrated AM-GM inequality $\dfrac{a + b}{2} \ge \sqrt{ab}$, which appears in every optimisation problem you will meet. Third, standard sums like $\sum k = \dfrac{n(n+1)}{2}$, $\sum k^2 = \dfrac{n(n+1)(2n+1)}{6}$, and $\sum k^3 = \left(\dfrac{n(n+1)}{2}\right)^2$.

The cleverest technique you will learn is the method of differences , when a series is neither AP nor GP, look at consecutive differences; sometimes those form an AP or GP and the original series can be summed indirectly.

For Class XII and beyond, sequences morph into infinite sums (series), and convergence becomes the question. For now, every series we write down is finite, which means every problem reduces to clean algebra.

What's inside

1. Sequences and notation , terms, $n$-th term, recurrences.
2. Arithmetic progression (AP) , definition, $n$-th term, sum, properties.
3. Geometric progression (GP) , definition, $n$-th term, sum, infinite GP.
4. Arithmetic and geometric means , single and multiple means, the AM-GM inequality.
5. Relationship between AP and GP, special sums , $\sum k$, $\sum k^2$, $\sum k^3$.
6. Method of differences and miscellaneous series.

Key results / Formula card

How to read this chapter

Memorise the three sum formulas above , they are nearly always needed. For every AP/GP problem, identify $a$, $d$ (or $r$), and $n$ before applying any formula. The AM-GM inequality looks like a curiosity but is your first systematic tool for proving inequalities and finding extrema without calculus.