Arithmetic Sequences

• An arithmetic sequence (or progression) is a sequence of numbers such that the difference between consecutive terms is constant. This constant value is known as the common difference, denoted by $d$.

• The first term of the sequence is typically denoted by $a$ or $u_1$. The position of any term in the sequence is represented by $n$, where $n$ is a positive integer ($n = 1, 2, 3, \dots$).

• If $d > 0$, the sequence is increasing; if $d < 0$, the sequence is decreasing; and if $d = 0$, the sequence is constant.

• The general term $u_n$ is defined by the starting value plus the common difference applied $(n-1)$ times.

• The arithmetic sequence is essentially a linear function restricted to integer domains. If you were to plot the terms $(n, u_n)$ on a coordinate plane, they would lie on a straight line with a slope equal to $d$.

• The relationship between any two terms $u_i$ and $u_j$ can be expressed as $u_j = u_i + (j-i)d$. This principle allows for the calculation of the common difference even if the first term is unknown.

• The arithmetic mean of two numbers is the value that, when placed between them, creates an arithmetic sequence. For any three consecutive terms $x, y, z$, the middle term $y = \frac{x+z}{2}$.

Finding the nth Term

• To find any specific term in the sequence, use the general formula:

General Term Formula: $u_n = a + (n-1)d$

• This formula accounts for the fact that the common difference is added one fewer time than the term's position (e.g., the 3rd term has 2 'jumps' from the 1st).

Solving for Unknowns

• If two non-consecutive terms are known, set up a system of linear equations or use the gradient-style formula: $d = \frac{u_j - u_i}{j - i}$.
• Once $d$ is found, substitute it back into the general formula with one of the known terms to solve for the first term $a$.

Summation (Arithmetic Series)

• The sum of the first $n$ terms can be calculated using two variations of the same logic:

1. If the last term ($L$) is known: $S_n = \frac{n}{2}(a + L)$
2. If only $a$ and $d$ are known: $S_n = \frac{n}{2}[2a + (n-1)d]$

• It is vital to distinguish between Arithmetic and Geometric sequences to apply the correct formulas during problem-solving.

• Sequence vs. Series: A sequence is the list of numbers ($u_1, u_2, u_3...$), while a series is the sum of those numbers ($u_1 + u_2 + u_3...$).

• The 'n' Check: Always ensure that $n$ is a positive integer. If a question asks 'Is 100 a term in this sequence?' and your calculation for $n$ results in a fraction, the answer is no.

• Sign Awareness: Be extremely careful with the sign of $d$. If the sequence is decreasing, $d$ must be negative. Forgetting the negative sign in the sum formula is a frequent cause of lost marks.

• Simultaneous Equations: When given two terms like $u_5$ and $u_{12}$, immediately write them as $a + 4d = X$ and $a + 11d = Y$. Subtracting these equations is the fastest way to eliminate $a$ and find $d$.

• Sanity Check: For summation problems, if $d$ is positive and $a$ is positive, $S_n$ must be larger than $u_n$. If your sum is smaller than your last term, re-check your arithmetic.