Sequences and Series

Flashcards covering definitions, formulas, and solving methods for arithmetic, geometric, and quadratic sequences and series as presented in the lecture notes.

What is a sequence, and what is its $n^{th}$ term denoted as?

A sequence is a list of numbers (e.g., $1, 2, 3, 4, 5$), and the $n^{th}$ term is denoted as $T_n$.

When is a sequence defined as arithmetic?

A sequence is arithmetic (or an Arithmetic Progression - AP) if the difference between each term and the next is constant: $T_{n+1} - T_n = d$, where $d$ is the common difference.

What is the formula for the $n^{th}$ term of an Arithmetic Progression (AP)?

$T_n = a + (n - 1)d$, where $a$ is the first term, $n$ is the number of the term, and $d$ is the common difference.

What is an arithmetic series?

An arithmetic series is the sum of an Arithmetic Progression (AP).

What are the two formulas for the sum of an arithmetic series ($S_n$)?

$S_n = \frac{n}{2}(2a + (n - 1)d)$ and $S_n = \frac{n(a+l)}{2}$.

When is a sequence defined as geometric?

A sequence is geometric (or a Geometric Progression - GP) if the ratio of each term to the previous term is constant: $\frac{T_{n+1}}{T_n} = r$, where $r$ is the common ratio.

What is the formula for the $n^{th}$ term of a geometric sequence?

$T_n = ar^{n-1}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the value of the term.

What mathematical method is required to find $n$ if it is the unknown in the $n^{th}$ term formula of a GP?

Exponential equations or logarithms must be used.

How many solutions exist for $r$ when solving $r^n = k$?

There are two solutions (one positive and one negative) when $n$ is even, and only one solution when $n$ is odd.

What is the formula for the sum of a geometric series ($S_n$)?

$S_n = \frac{a(r^n - 1)}{r - 1}$.

Under what condition does the sum of a geometric series reach a limit (converge)?

The series reaches a limit when $-1 < r < 1$; in this case, each subsequent term gets closer to zero.

What is the formula for the limiting sum ($S_{\infty}$) of a geometric series?

$S_{\infty} = \frac{a}{1 - r}$.

In the general formula for a quadratic sequence ($T_n = an^2 + bn + c$), how is the value of $a$ determined?

$a$ is calculated as half of the second difference (e.g., if the second difference is $4$, then $a = \frac{4}{2} = 2$).

What remains of a quadratic sequence formula after subtracting $an^2$ from each term?

A linear sequence (arithmetic sequence) of the form $bn + c$ remains.

What are the common differences for the arithmetic sequences $18, 22, 26, 30, \text{...}$ and $15, 8, 1, -6, \text{...}$?

The common difference is $+4$ in the first sequence and $-7$ in the second.