Sequences and Series Class 11: notes and important questions

Sequences and Series: revision notes

1. Sequences and series

A sequence is an ordered list \(a_1, a_2, a_3, \ldots\), often given by a formula for \(a_n\) or by a recurrence. A series is the sum of its terms, \(a_1 + a_2 + \cdots\); \(S_n\) is the sum of the first \(n\) terms.

2. Arithmetic mean

The AM of \(a\) and \(b\) is \(A = \dfrac{a + b}{2}\), so \(a, A, b\) are in AP. To insert \(n\) AMs between \(a\) and \(b\), use common difference \(d = \dfrac{b - a}{n + 1}\).

3. Geometric progressions

A GP multiplies by a fixed non-zero ratio \(r\): \(a, ar, ar^2, \ldots\)

• General term: \(a_n = ar^{n-1}\).
• Sum of \(n\) terms: \(S_n = \dfrac{a(r^n - 1)}{r - 1} = \dfrac{a(1 - r^n)}{1 - r}\) for \(r \neq 1\); \(S_n = na\) if \(r = 1\).
• Three terms in GP: \(\dfrac ar, a, ar\) (their product is \(a^3\)).

4. Infinite GP

If \(|r| \lt 1\), \(r^n \to 0\) and \(S_\infty = \dfrac{a}{1 - r}\). Recurring decimals are infinite GPs: \(0.\overline{36} = \dfrac{36}{100} + \dfrac{36}{100^2} + \cdots = \dfrac{36}{99} = \dfrac{4}{11}\).

5. Geometric mean and AM ≥ GM

The GM of positive \(a, b\) is \(G = \sqrt{ab}\). To insert \(n\) GMs between \(a\) and \(b\), use \(r = \left(\dfrac ba\right)^{\frac{1}{n + 1}}\). For positive \(a, b\): \(A - G = \dfrac{(\sqrt a - \sqrt b)^2}{2} \ge 0\), so \(A \ge G\), with equality only when \(a = b\). If \(A\) and \(G\) are known, \(a, b\) are the roots of \(x^2 - 2Ax + G^2 = 0\).

Worked example 1

Find the sum of the first \(7\) terms of \(2, 6, 18, \ldots\)

\(S_7 = \dfrac{2(3^7 - 1)}{3 - 1} = 3^7 - 1 = 2186\).

Worked example 2

Find \(1 - \dfrac12 + \dfrac14 - \cdots\) to infinity.

\(r = -\dfrac12\): \(S_\infty = \dfrac{1}{1 + \frac12} = \dfrac23\).

Worked example 3

Insert two GMs between \(2\) and \(54\).

\(54 = 2r^3 \Rightarrow r = 3\): \(6, 18\).

Common errors

• \(a_n = ar^n\) instead of \(ar^{n-1}\).
• Using \(S_\infty\) when \(|r| \ge 1\): the sum does not exist.
• Taking the ratio as \(\dfrac{a_1}{a_2}\) instead of \(\dfrac{a_2}{a_1}\), especially with negative ratios.
• Forgetting that the GM needs positive numbers.

Exam tips

• When two terms of a GP are given, divide the equations to find \(r\) first.
• In AM–GM questions, state "for positive numbers" before using \(A \ge G\).