Sequences & Series

Sequences & Series


Sequences & Series cover Arithmetic, Geometric and Harmonic Progressions. This post will enumerate the essential formulae with a link to problems and solutions thereof. Study Complex Numbers.

What is a Sequence in mathematics?

A sequence is an ordered collection of numbers \(t_1, t_2, t_3, … , t_n, …\). The word of importance is ‘ordered’. It signifies a recognizable definite rule of relationship between the terms or elements. Following the rule, the general term \(t_n\) is determinable for any value of the natural number \(n\). A sequence can be finite or infinite.

Arithmetic Progression

If the difference between any term and the preceding term of a sequence is constant, it is in arithmetic progression. Symbolically, \[t_n – t_n-1 = k\]

General Form \[ a + (a + d) + (a + 2d) + (a + 3d) + ⋯ +\{a + (n – 1)d\} \]
How to find out a particular term? \[ t_n = a + (n – 1)d \]
Sum of all the terms or up to a particular point \[ S_n = \frac{n}{2}(a + l) = \frac{n}{2}\{2a + (n – 1)d\} \]
Find the Last Term itself \[ l = a + (n – 1)d \]
You are familiar with Arithmetic Mean since long \[ \frac{a + b}{2} \]
So, let’s insert ‘n’ A.M’s between ‘a’ and ‘b’: \[ \frac{an + b}{n + 1}, \frac{a(n – 1) + 2b}{n + 1}, \frac{a(n – 2) + 3b}{n + 1}, … , \frac{a + nb}{n + 1} \]
Can you find sum of the ‘n’ A.M’s? Yes. \[ A_1 + A_2 + A_3 + ⋯ +A_n = \frac{a + b}{2}.n \]

Tricks

Take, when needed, as three consecutive terms of an A.P. \[ (\alpha – \beta), \alpha , (\alpha + \beta) \]
And, when four is for asking: \[ (\alpha – 3\beta), (\alpha – \beta), (\alpha + \beta), (\alpha + 3\beta) \]

Geometric Progression

G.P or G.S is a sequence in which the ratio of any term and the preceding term is always a constant. Symbolically, \[ \frac{t_n}{t_{n – 1}} = k, \text{ for }n \geq 2 \]

Properties:

  • G.S is completely known if one term and the common ratio \(r\) are known.
  • Multiply/divide all terms with the same quantity, we get a new G.S.
  • The product of the corresponding terms of two G.S results in a new G.S.
  • Reciprocals of all the terms make a new G.S.

Symbolically, Let given G.S be: \[ a + ar + ar^2 + ar^3 + ⋯ +ar^{n – 1} \] Then, these are also in G.S: \[ ab + abr + abr^2 + ⋯ +abr^{n – 1},\text{ with }t_1 = ab\text{ and }\text{c.r. } = r \] \[ \frac{a}{b} + \frac{ar}{b} + \frac{ar^2}{b} + ⋯ +\frac{ar^{n – 1}}{b},\text{ with }t_1 = \frac{a}{b}\text{ and c.r. } = r \] \[ \frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \frac{1}{ar^3} + ⋯, \text{ with }t_1 = \frac{1}{a}\text{ and c.r. } = \frac{1}{r} \]

Formulae

\[ t_n = ar^{n – 1} \] \[ S_n = \frac{a(1 – r^n)}{1 – r} \] \[ S_{\propto } = \frac{a}{1 – r}, \text{ where }|r| < 1 \] \[ \text{ G.M } = \sqrt{ab} \]

Three consecutive terms: \[\frac{a}{r}, a, ar \]

Four consecutive terms: \[ \frac{a}{r^3}, \frac{a}{r}, ar, ar^3 \]

Harmonic Progression

Harmonic series is formed by the reciprocals of an arithmetic series. General form: \[ \frac{1}{a} + \frac{1}{a + d} + \frac{1}{a + 2d} + ⋯ \] To find \(t_n\) of H.P, we follow the A.P route. No general formula to sum up an H.P.
Harmonic Mean and relations between respective means: A, G and H are in G.P. \[ H.M. = \frac{2ab}{a + b}, \text{ where } a \neq -b\] \[ A \times H = G^2 \] \[ A > G > H \]

Wikipedia

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