What does the variable $n$ represent in an nth term formula?

The variable $n$ represents the position or 'rank' of the term in the sequence (e.g., $n=5$ for the fifth term).

What is the general algebraic form of a quadratic sequence?

The general form is $u_n = an^2 + bn + c$, where $a, b,$ and $c$ are constants and $a \neq 0$.

What is the fundamental difference between a linear sequence and a quadratic sequence regarding their differences?

A linear sequence has a constant first difference between terms. A quadratic sequence has a changing first difference, but a constant second difference (the difference between the differences).

How does the growth of a linear sequence compare to a quadratic sequence as $n$ becomes very large?

A linear sequence grows at a constant rate, forming a straight line if graphed. A quadratic sequence grows at an increasing (or decreasing) rate, forming a parabolic curve, and will eventually grow much faster than any linear sequence.

When solving for 'which term' has a certain value, what must be true about the resulting value of $n$?

The value of $n$ must be a positive integer (1, 2, 3...). If the calculation results in a fraction, decimal, or negative number, that specific value does not exist as a term in the sequence.

What is a common mistake when identifying the coefficient $a$ in the quadratic formula $an^2 + bn + c$?

Students often mistakenly use the second difference itself as the coefficient $a$. The correct coefficient $a$ is always half of the second difference ($a = \frac{d_2}{2}$).

Why is it incorrect to use $n=0$ when finding the first term of a sequence?

By mathematical convention, sequences start at the first position, which is $n=1$. Using $n=0$ would refer to a 'zeroth' term, which does not exist in standard sequence notation.

What error occurs if you only check the first term when verifying an nth term formula?

Many different formulas can produce the same first term. You must check at least the first two or three terms to ensure the formula correctly captures the pattern of change (the difference) between terms.

Define the 'common difference' in the context of an arithmetic sequence.

The common difference is the constant value added to (or subtracted from) each term to get the next term in a linear sequence.

Why does a constant second difference imply that the formula involves $n^2$?

In mathematics, the second difference is analogous to the second derivative in calculus. Since the second derivative of a quadratic function is a constant, a constant second difference in a discrete sequence indicates a quadratic relationship.

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2. Algebra & Functions

Finding nth Terms of Sequences

Summary

The nth term of a sequence is an algebraic rule that defines the value of a term based on its position, $n$ , in the list. By identifying patterns in the differences between terms, one can determine if a sequence is linear or quadratic and derive a general formula to calculate any term or identify the position of a specific value.

1. Definition & Core Concepts

A sequence is an ordered list of numbers where each number is called a term. The position of a term in the sequence is denoted by the variable $n$ , which must always be a positive integer ( $n = 1, 2, 3, ...$ ).
The nth term (often written as $u_n$ or $a_n$ ) is a general formula that generates the value of any term in the sequence when the position $n$ is substituted into the expression.
Finding the nth term involves identifying the mathematical relationship between the position $n$ and the value of the term at that position.

2. Linear Sequences (Arithmetic)

A graph showing a linear sequence where the points form a straight line, illustrating the constant common difference 'd' between terms.

3. Quadratic Sequences

4. Methodology: Finding the Quadratic Formula

5. Key Distinctions

6. Exam Strategy & Tips