Sequences and Series - Theory & Concepts

Sequences and Series

A sequence is an ordered list of numbers, and a series is the sum of those numbers. These mathematical constructs form the foundation of engineering economics (calculating compound interest, gradients) and numerical methods.

Arithmetic Progressions (AP)

An arithmetic progression is a sequence where the difference between consecutive terms is constant. This constant is called the common difference (

d

AP Formulas

$n$ -th Term Formula: $a_n = a_1 + (n - 1)d$
Sum of First $n$ Terms ( $S_n$ ): The sum of an AP is the average of the first and last terms, multiplied by the number of terms.

S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]

Geometric Progressions (GP)

A geometric progression is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (

r

GP Formulas

$n$ -th Term Formula: $a_n = a_1 r^{n-1}$
Sum of First $n$ Terms ( $S_n$ ): For $r \neq 1$ , $S_n = a_1 \frac{1 - r^n}{1 - r}$

Infinite Geometric Series

If the absolute value of the common ratio is less than 1 (

|r| \lt 1

), the sum of an infinite geometric progression converges to a finite value.

Infinite Sum Formula

Calculates the sum of an infinite geometric series that converges.

S_{\infty} = \frac{a_1}{1 - r} \quad (|r| \lt 1)

Variables

Symbol	Description	Unit
$S_{\infty}$	The sum of the infinite series	-
$a_1$	The first term of the series	-
$r$	The common ratio (must be strictly between -1 and 1)	-

Interactive Sequence Generator

Experiment with the starting value, difference/ratio, and sequence type to see how the terms grow and their sum accumulates.

Sequence & Series Explorer

Starting Value (a₁)2

Common Difference (d)3

Number of Terms (n)10

Term Value (aₙ)

a_{10} = 29

Series Sum (Sₙ)

S_{10} = 155

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Harmonic Progressions (HP)

A sequence is a harmonic progression if the reciprocals of its terms form an arithmetic progression. It is useful in calculations involving rates or parallel circuits.

HP Properties

If the terms $a_1, a_2, a_3 \dots$ form an HP, then $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3} \dots$ form an AP.
To solve HP problems, invert all terms, use AP formulas, and invert the final answer.

AM-GM-HM Inequality

For any set of positive numbers, their Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) are related by:

\text{AM} \ge \text{GM} \ge \text{HM}

Relationship: For two positive numbers $a$ and $b$ , $\text{GM}^2 = \text{AM} \times \text{HM}$ .

Mathematical Induction

Mathematical induction is a rigorous logical method used to prove that a statement or formula is true for all natural numbers (

n = 1, 2, 3, \dots

). It operates like a line of falling dominoes: if you can push the first one over, and you know that any falling domino will push over the next one, then all dominoes will fall.

The Principle of Mathematical Induction

STEP-BY-STEP

Base Case: Verify that the statement is true for the first possible value of $n$ (usually $n = 1$ ). If the base case is false, the induction fails immediately.
Inductive Hypothesis: Assume the statement is true for an arbitrary integer $n = k$ , where $k \ge 1$ . Write down the formula exactly as given, substituting $k$ for $n$ .
Inductive Step: Using the assumption from the Inductive Hypothesis, logically prove that the statement must also be true for the next integer, $n = k + 1$ . Substitute $k + 1$ into the original formula to see what you are trying to reach.
Conclusion: If both the base case and the inductive step are proven, then by the Principle of Mathematical Induction, the statement holds for all natural numbers $n$ .

The Binomial Theorem

The Binomial Theorem provides a quick way to expand binomial expressions

(x + y)^n

without having to multiply out the terms manually.

Binomial Expansion

The coefficients of the expansion are found using Pascal's Triangle or combinations (nCr).

(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k

\binom{n}{k} = \frac{n!}{k!(n-k)!}

Key Takeaways

Arithmetic vs Geometric: APs grow by adding a constant difference ( $d$ ). GPs grow by multiplying by a constant ratio ( $r$ ).
Convergence: An infinite series only has a finite sum if it is a geometric series with $|r| \lt 1$ .
Binomial Symmetry: The coefficients of a binomial expansion are symmetric and match the rows of Pascal's Triangle.
k-value in Binomials: Remember that the formula for the $(k+1)$ th term uses $k$ , not $k+1$ .
Mathematical Induction: A two-step logical proof technique used to establish statements for all natural numbers.

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