Number and Algebra - Sequences and Series (Arithmetic & Geometric)

Arithmetic Sequences (Linear Growth): An arithmetic sequence is a pattern where the difference between consecutive terms is constant, known as the common difference $d$. Visually, if you plot the term number $n$ on the x-axis and the term value $u_n$ on the y-axis, the points will form a discrete straight line. The gradient of this line corresponds to $d$, and the sequence is increasing if $d > 0$ and decreasing if $d < 0$.

Geometric Sequences (Exponential Change): In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio $r$. This results in exponential growth if $r > 1$ or exponential decay if $0 < r < 1$. Visually, these sequences form a curve that either accelerates away from the x-axis or flattens out towards it. If $r$ is negative, the terms will alternate between positive and negative values, creating a zig-zag visual pattern above and below the horizontal axis.

Sigma Notation (Summation): The symbol $\sum$ is used to represent the sum of a sequence of numbers. The expression $\sum_{k=1}^{n} u_k$ indicates that you should substitute integers from the lower limit $1$ to the upper limit $n$ into the general term $u_k$ and add the results together. Visually, this can be represented as the total area of a series of rectangles where the width is 1 and the height is the value of each term.

Convergence of Geometric Series: A geometric series is said to converge if the sum of its terms approaches a finite limit as the number of terms goes to infinity. This only occurs when the absolute value of the common ratio is less than one ($|r| < 1$). On a graph of the partial sums $S_n$ vs $n$, a convergent series will appear to level off toward a horizontal asymptote, which represents the sum to infinity $S_{\infty}$.

Arithmetic Series and Partial Sums: An arithmetic series is the sum of the terms in an arithmetic sequence. The formula for the sum $S_n = \frac{n}{2}(u_1 + u_n)$ can be visualized as finding the area of a trapezoid, where $n$ is the height and the two parallel sides are the first and last terms. As $n$ increases, the sum of an arithmetic sequence grows quadratically.

Applications to Finance: Many financial models are based on sequences. Simple interest and linear depreciation follow arithmetic patterns (linear growth/decay). In contrast, compound interest and reducing balance depreciation follow geometric patterns (exponential growth/decay), where the value changes by a fixed percentage each period. This can be visualized by comparing a straight line (simple interest) to a curve that grows steeper over time (compound interest).