Arithmetic Sequences

• An arithmetic sequence is an ordered list of numbers where the difference between any two consecutive terms is constant. This consistent pattern makes the sequence predictable and allows for the derivation of general formulas.

• The constant difference between successive terms is called the common difference, denoted by $d$. This value can be positive, indicating an increasing sequence, or negative, indicating a decreasing sequence.

• Each number in the sequence is referred to as a term, and its position is denoted by $n$ (e.g., $n=1$ for the first term, $n=2$ for the second term). The first term of an arithmetic sequence is typically denoted by $a$ or $u_1$.

• Arithmetic sequences are also frequently called linear sequences because when the term values are plotted against their positions, the resulting points form a straight line. This linearity is a key characteristic that connects them to linear functions.

• The linear nature of arithmetic sequences stems directly from the constant addition or subtraction of the common difference. Each increment in term position ($n$) corresponds to a fixed change in the term's value ($u_n$).

• This relationship is analogous to a linear function $y = mx + c$, where the common difference $d$ acts as the gradient (slope) $m$, and the first term $a$ influences the y-intercept. Specifically, $u_n$ can be seen as a linear function of $n$.

• The formula for the $n$-th term, $u_n = a + (n-1)d$, explicitly shows this linearity. As $n$ increases by 1, $u_n$ changes by exactly $d$, demonstrating a constant rate of change across the sequence.

• The general formula for the $n$-th term of an arithmetic sequence is given by:

$u_n = a + (n - 1)d$

• In this formula, $u_n$ represents the value of the term at position $n$. The variable $a$ denotes the first term of the sequence, which is the value when $n=1$.

• The variable $d$ is the common difference, the constant value added to each term to get the next. The term $(n-1)d$ accounts for the fact that to reach the $n$-th term from the first term, the common difference is applied $n-1$ times.

• This formula is crucial because it allows for the direct calculation of any term in the sequence without needing to list all preceding terms. For example, to find the 50th term, one simply substitutes $n=50$ into the formula.

• Finding a specific term: If the first term ($a$), common difference ($d$), and the desired term position ($n$) are known, simply substitute these values into the $n$-th term formula $u_n = a + (n-1)d$ to calculate $u_n$. For instance, if $a=5$, $d=3$, and you want the 7th term, $u_7 = 5 + (7-1)3 = 5 + 6 \times 3 = 23$.

• Deriving the general formula: To find an expression for $u_n$ in terms of $n$, substitute the known values of $a$ and $d$ into the formula and simplify the algebraic expression. This results in a linear equation in $n$, such as $u_n = 3n + 2$.

• Finding $a$ and $d$ from given terms: When two non-consecutive terms of an arithmetic sequence are provided, such as $u_x$ and $u_y$, two simultaneous linear equations can be formed using the $n$-th term formula. For example, $u_x = a + (x-1)d$ and $u_y = a + (y-1)d$. Solving these equations will yield the values for $a$ and $d$.

• Checking if a number is in the sequence: To determine if a particular value belongs to an arithmetic sequence, set the value equal to the $n$-th term formula and solve for $n$. If $n$ is a positive integer, the number is in the sequence; otherwise, it is not.

• Arithmetic vs. Geometric Sequences: The primary distinction lies in their progression rule. Arithmetic sequences involve a common difference (addition/subtraction), leading to linear growth or decay. Geometric sequences, conversely, involve a common ratio (multiplication/division), resulting in exponential growth or decay.

• Arithmetic Sequences vs. General Sequences: While an arithmetic sequence is a type of sequence, not all sequences are arithmetic. A general sequence can follow any rule (e.g., Fibonacci sequence, quadratic sequences), whereas an arithmetic sequence is strictly defined by its constant common difference. This specific rule makes arithmetic sequences predictable and easily formulable.

• Memorize the Formula: The $n$-th term formula for an arithmetic sequence, $u_n = a + (n-1)d$, is fundamental and often not provided in formula booklets. Ensure you know it by heart and understand what each variable represents.

• Identify $a$ and $d$ Clearly: Before attempting any calculations, always explicitly write down the first term ($a$) and the common difference ($d$). Pay close attention to the sign of $d$, especially for decreasing sequences.

• Simultaneous Equations for Unknowns: If you are given two terms of a sequence and asked to find $a$ and $d$, immediately think of setting up two linear equations using the $n$-th term formula. This is a common problem type that tests algebraic proficiency.

• Verify $n$ is a Positive Integer: When determining if a specific number is part of a sequence, solve for $n$. If $n$ is not a positive whole number (e.g., a fraction, decimal, or negative), then the number is not a term in the sequence. This is a critical check for validity.

• Incorrect Common Difference Calculation: A frequent error is calculating $d$ as $u_n - u_{n+1}$ instead of $u_{n+1} - u_n$. Always subtract a term from the term that immediately follows it to get the correct sign for $d$. For example, in $10, 7, 4$, $d = 7 - 10 = -3$, not $10 - 7 = 3$.

• Misapplication of $(n-1)$: Students sometimes forget the $(n-1)$ factor in the formula, incorrectly using $u_n = a + nd$. This leads to an offset in the sequence, as the first term would then be $a+d$ instead of $a$. Remember that $d$ is added $n-1$ times to the first term.

• Algebraic Errors in Simultaneous Equations: When solving for $a$ and $d$ from two given terms, algebraic mistakes during substitution or elimination are common. Double-check calculations, especially when dealing with negative numbers or fractions.

• Assuming $n$ can be non-integer: The term position $n$ must always be a positive integer ($1, 2, 3, ...$). If solving for $n$ yields a non-integer or negative value, it means the number in question is not a valid term in the sequence.