Sequences and Progressions Class 9: notes and important questions

Sequences and Progressions: revision notes

1. Sequences and their rules

A sequence is a list of numbers in a definite order: \(a_1, a_2, a_3, \ldots\). It can be given by

• an explicit rule for the \(n\)th term, such as \(a_n = n^2 + 1\) (gives any term directly), or
• a recursive rule: the first term(s) and how each term comes from the ones before, such as \(a_1 = 2,\ a_{n+1} = 3a_n - 1\).

2. The Virahanka-Fibonacci sequence

\(1, 1, 2, 3, 5, 8, 13, \ldots\): each term after the second is the sum of the two before, \(F_{n+1} = F_n + F_{n-1}\). Virahanka found these numbers by counting the rhythms of poetic metres (in 1 and 2 beats) centuries before Fibonacci.

3. Arithmetic progressions

An AP adds the same number \(d\) (the common difference) each time: \(a, a + d, a + 2d, \ldots\) with

\[a_n = a + (n - 1)d\]

Recursively, \(a_{n+1} = a_n + d\). Its graph (term against \(n\)) is a set of points on a straight line: linear growth.

4. Sum of the first \(n\) natural numbers

\[1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}\]

Write the sum forwards and backwards: each of the \(n\) columns adds to \(n + 1\), so twice the sum is \(n(n + 1)\). Also \(1 + 3 + 5 + \cdots + (2n - 1) = n^2\).

5. Geometric progressions

A GP multiplies by the same number \(r\) (the common ratio) each time: \(a, ar, ar^2, \ldots\) with \(a_n = ar^{n-1}\). Recursively, \(a_{n+1} = r\,a_n\). Doubling, halving and repeated percentage change are GPs.

6. Patterns from repeated processes

• Tower of Hanoi: the least number of moves for \(n\) discs satisfies \(M_{n} = 2M_{n-1} + 1\), \(M_1 = 1\), so \(M_n = 2^n - 1\).
• Fractals: each stage repeats a rule on every piece, so counts and areas often form GPs (for example \(3^{n-1}\) triangles, each stage keeping \(\dfrac34\) of the shaded area).

Worked example 1

Which term of the AP \(4, 11, 18, \ldots\) is \(144\)?

\(4 + 7(n - 1) = 144 \Rightarrow n - 1 = 20 \Rightarrow n = 21\).

Worked example 2

Find the \(6\)th term of the GP \(2, 6, 18, \ldots\).

\(r = 3\): \(a_6 = 2 \times 3^5 = 486\).

Worked example 3

Find \(1 + 2 + \cdots + 40\).

\(\dfrac{40 \times 41}{2} = 820\).

Common errors

• \(a_n = a + nd\) instead of \(a + (n - 1)d\); similarly \(ar^n\) instead of \(ar^{n-1}\).
• Calling a sequence an AP from its first two terms only: check that every difference is the same.
• Ratio taken as \(\dfrac{a_1}{a_2}\) instead of \(\dfrac{a_2}{a_1}\).
• Accepting a non-integer \(n\) when asked "which term": then the number is not a term.

Exam tips

• State \(a\) and \(d\) (or \(r\)) at the start of every progression question.
• For recursive rules, write each term on its own line: \(a_2 = \ldots\), \(a_3 = \ldots\).