Arithmetic Sequence

The Core Identity

An Arithmetic Sequence is a list of numbers where each term is found by adding a constant common difference to the previous one, creating linear growth.

$$d = a_{n} - a_{n-1}$$

The common difference is the gap between any two consecutive terms.

$d$=Common difference(unitless)

$a_{n}$=Any term in the sequence(unitless)

$a_{n-1}$=The term immediately preceding $a_{n}$(unitless)

$d > 0$

→

The sequence is increasing.

$d < 0$

→

The sequence is decreasing.

$d = 0$

→

All terms are equal (constant sequence).

Linear Growth: Because the change is constant, the terms follow a straight-line pattern when plotted against index $n$.

Consecutive Check: The difference must be identical for ALL pairs of neighbors — check at least two pairs to confirm an A.P.

Non-Consecutive Shortcut: For any two terms $a_m$ and $a_p$, the common difference is $d = \frac{a_p - a_m}{p - m}$.

The General Term

The general term allows you to calculate the value of any specific index without listing every intermediate number.

$$a_n = a_1 + (n - 1)d$$

The n-th term is the starting value plus $(n-1)$ jumps of size $d$.

$a_n$=Value of the n-th term(unitless)

$a_1$=First term (starting value)(unitless)

$n$=Position (index) of the term(positive integer)

$d$=Common difference(unitless)

$n = 1$

→

$a_1 = a_1 + 0 \cdot d = a_1$ (zero jumps from the start).

Linear Function Form: $a_n = dn + (a_1 - d)$ — this is a linear function of $n$ with slope $d$ and y-intercept $(a_1 - d)$.

Finding $n$ for a Given Term: Rearrange to $n = \frac{a_n - a_1}{d} + 1$. If this isn't a positive integer, the value is not in the sequence.

Shifted Index Problems: If given $a_{n-k} = f(n)$, substitute values of $n$ that make the subscript equal to 1, 2, 3... to find the actual first terms.

Arithmetic Series (Sum of Terms)

An Arithmetic series is the partial sum of the terms in a sequence. When you know the first and last terms, use Gauss's pairing method.

$$S_n = \frac{n}{2}(a_1 + a_n)$$

The sum equals the average of the first and last terms, multiplied by the number of terms.

$S_n$=Sum of the first $n$ terms(unitless)

$n$=Number of terms to sum(positive integer)

$a_1$=First term(unitless)

$a_n$=Last term in the sum(unitless)

Gauss's Insight: Pair the first term with the last, the second with the second-last, etc. Each pair sums to $a_1 + a_n$, and there are $\frac{n}{2}$ such pairs.

When to Use: Choose this form when you already know (or can easily find) $a_n$.

When the last term $a_n$ is unknown, substitute $a_n = a_1 + (n-1)d$ into the quick formula to get the standard form.

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

Expresses the sum entirely in terms of $a_1$, $d$, and $n$ — no need to find $a_n$ first.

$S_n$=Sum of the first $n$ terms(unitless)

$a_1$=First term(unitless)

$d$=Common difference(unitless)

$n$=Number of terms(positive integer)

When to Use: Choose this form when you know $a_1$ and $d$ but not $a_n$.

Quadratic in $n$: $S_n = \frac{d}{2}n^2 + (a_1 - \frac{d}{2})n$ — the sum of an A.P. is always a quadratic function of $n$.

The Arithmetic Mean

The arithmetic mean is the midpoint between two numbers; when inserted between them, it creates a three-term arithmetic sequence.

$$A = \frac{a + b}{2}$$

The A.M. of two numbers is their average — equidistant from both.

$A$=Arithmetic mean(unitless)

$a, b$=The two endpoint numbers(unitless)

Equidistant Property: $A - a = b - A = d$, so the common difference is $d = \frac{b - a}{2}$.

General A.M. in a Sequence: Any term $a_n$ is the arithmetic mean of its two neighbours: $a_n = \frac{a_{n-1} + a_{n+1}}{2}$.

To insert $n$ arithmetic means between two numbers $a$ and $b$, construct an A.P. of $(n+2)$ terms with $a$ as the first term and $b$ as the last.

$$d = \frac{b - a}{n + 1}$$

The $(n+2)$-term A.P. has $(n+1)$ gaps of equal size between $a$ and $b$.

$d$=Common difference of the extended A.P.(unitless)

$a$=First endpoint (= $a_1$)(unitless)

$b$=Last endpoint (= $a_{n+2}$)(unitless)

$n$=Number of means to insert(positive integer)

Denominator is $n+1$: With $n$ means, there are $n + 2$ total terms and $n + 1$ gaps.

The $k$-th Mean: $A_k = a + k \cdot \frac{b-a}{n+1}$ for $k = 1, 2, \ldots, n$.

Sum of $n$ AMs: The sum of all $n$ arithmetic means between $a$ and $b$ equals $n \times \frac{a+b}{2}$ (i.e., $n$ times their A.M.).