Arithmetic Sequences - Properties and Applications

A sequence is simply an ordered, numbered list of terms, $u_1, u_2, u_3, \ldots$ — a function whose domain is the positive integers rather than the reals. The simplest, most common pattern a sequence can follow is that each term is obtained from the one before it by adding a fixed amount: $5, 8, 11, 14, 17, \ldots$. This chapter studies exactly that pattern, the arithmetic sequence, builds the formula for its general term and the sum of its terms, and applies both to realistic problems. The tools developed here — a recursive description versus an explicit formula, and the derivation of a sum formula by a clever pairing trick — reappear with only minor changes when geometric sequences are studied in the next chapter.

Have to hand: comfort with solving simultaneous linear equations, since many applied problems give two pieces of information about a sequence and ask you to recover its defining constants.

1. Defining an Arithmetic Sequence

An arithmetic sequence is a sequence in which the difference between any term and the one immediately before it is constant. This constant is called the common difference, denoted $d$:

Equivalently, an arithmetic sequence can be defined recursively: $u_1$ is given, and $u_{n+1} = u_n + d$ for $n \geq 1$. A recursive definition is easy to state but inconvenient to use directly — finding $u_{50}$ from a recursive definition requires computing all 49 preceding terms.

Recursive versus explicit. A recursive formula defines each term from the previous one(s); an explicit (or closed) formula gives $u_n$ directly from $n$, with no reference to other terms. Converting from recursive to explicit is one of the central tasks of this chapter.

2. The General Term Formula

Starting from $u_1$ and adding $d$ repeatedly:

the pattern is that $u_n$ is reached from $u_1$ by adding $d$ exactly $n-1$ times (not $n$ times — a frequent source of off-by-one errors):

This can be proved rigorously by induction (Chapter 3): the base case $n=1$ gives $u_1 = u_1 + 0 = u_1$, true; the inductive step assumes $u_k = u_1+(k-1)d$ and uses the recursive definition $u_{k+1}=u_k+d$ to get $u_{k+1} = u_1+(k-1)d+d = u_1+kd = u_1+((k+1)-1)d$, matching the formula at $n=k+1$.

Worked example. An arithmetic sequence has first term $u_1 = 7$ and common difference $d=-3$. Find $u_{20}$.

Worked example — finding $u_1$ and $d$ from two terms. In an arithmetic sequence, $u_5 = 23$ and $u_{12} = 58$. Find $u_1$, $d$, and $u_{30}$.

Using the general term formula for each given term:

Subtracting the first equation from the second eliminates $u_1$:

Substituting back: $u_1 + 4(5) = 23 \Rightarrow u_1 = 3$. Then $u_{30} = 3 + 29(5) = 3+145=148$.

The subtraction trick. Whenever two terms $u_m$ and $u_n$ of an arithmetic sequence are known, subtracting their general-term equations eliminates $u_1$ immediately, leaving $d = \dfrac{u_n-u_m}{n-m}$ — faster than solving the full simultaneous system from scratch.

3. The Arithmetic Mean

If $a$, $x$, $b$ are three consecutive terms of an arithmetic sequence, then $x-a = b-x$ (the common difference is the same on both sides of $x$), so

The middle term $x$ is called the arithmetic mean of $a$ and $b$ — literally their average. More generally, to insert $k$ arithmetic means between two numbers $a$ and $b$ (creating an arithmetic sequence of $k+2$ terms with $a$ first and $b$ last), treat $a$ as $u_1$ and $b$ as $u_{k+2}$, and solve $b = a + (k+1)d$ for the common difference $d = \dfrac{b-a}{k+1}$.

Worked example. Insert three arithmetic means between $5$ and $25$. Here $a=5$, $b=25=u_5$ (three means plus the two endpoints make five terms), so $d = \dfrac{25-5}{4} = 5$. The sequence is $5, 10, 15, 20, 25$.

4. The Sum of an Arithmetic Series

A series is the sum of the terms of a sequence. Denote the sum of the first $n$ terms of an arithmetic sequence by $S_n$. Writing the sum forwards and backwards and adding the two (a method attributed to the young Gauss):

Adding these two equations term by term, each column sums to $u_1+u_n$ (since whatever is added to $u_1$ in the top row is subtracted from $u_n$ in the bottom row), and there are $n$ columns:

Substituting $u_n = u_1+(n-1)d$ gives the equivalent form in terms of $u_1$ and $d$ alone, useful when $u_n$ is not already known:

Worked example. Find the sum of the first $30$ terms of the arithmetic sequence with $u_1=4$, $d=6$.

Worked example — a classic Gauss-style sum. Find $7 + 14 + 21 + \cdots + 700$ (the sum of the first however-many multiples of $7$ up to $700$). First identify $n$: this is an arithmetic sequence with $u_1=7$, $d=7$, $u_n=700$, so

Then $S_{100} = \dfrac{100}{2}(7+700) = 50(707) = 35\,350$.

5. Recovering a Term from $S_n$ and $S_{n-1}$

Since $S_n = u_1+u_2+\cdots+u_n$ and $S_{n-1}=u_1+\cdots+u_{n-1}$, subtracting recovers the individual term:

This relation holds for any sequence, not only arithmetic ones, and is useful whenever a problem gives a formula for $S_n$ directly (rather than for $u_n$) and asks for individual terms.

Worked example. A sequence has $S_n = 3n^2+2n$. Find $u_1$ and $u_5$, and determine whether the sequence is arithmetic.

$u_1 = S_1 = 3(1)^2+2(1)=5$. For $n\geq2$: $u_n = S_n-S_{n-1} = \big(3n^2+2n\big)-\big(3(n-1)^2+2(n-1)\big)$. Expanding $3(n-1)^2+2(n-1) = 3n^2-6n+3+2n-2=3n^2-4n+1$, so

Checking $n=1$ in this formula: $6(1)-1=5=u_1$, so in fact $u_n=6n-1$ holds for all $n\geq1$ here (not guaranteed in general — always check). Since $u_n=6n-1$ is linear in $n$, the sequence is arithmetic with $d=6$; indeed $u_5=6(5)-1=29$.

6. Applications of Arithmetic Sequences

Arithmetic sequences model any situation with constant additive change per step: seating in rows of a theatre with a constant increase per row, a salary increasing by a fixed amount each year, simple (non-compound) interest, depreciation by a fixed amount per year, stacking objects in a triangular pile.

Worked example. A theatre has $20$ rows. The first row has $18$ seats, and each subsequent row has $2$ more seats than the row before it. Find the number of seats in row $20$, and the theatre's total capacity.

This is arithmetic with $u_1=18$, $d=2$, $n=20$:

7. Common Exam Pitfalls

• Using $nd$ instead of $(n-1)d$. The general term adds $d$ exactly $n-1$ times from $u_1$, not $n$ times — check with $n=1$: $u_1=u_1+(1-1)d=u_1$, correctly adding nothing.
• Confusing $S_n$ (sum of first $n$ terms) with $u_n$ (the $n$th term itself). A question asking for "the $20$th term" wants $u_{20}$; one asking for "the sum of the first $20$ terms" wants $S_{20}$ — mixing these up is extremely common under time pressure.
• Forgetting $u_1=S_1$ is a special case when recovering terms from a sum formula — the relation $u_n=S_n-S_{n-1}$ only applies for $n\geq2$.
• Miscounting the number of terms when inserting arithmetic means. Inserting $k$ means between $a$ and $b$ creates $k+2$ terms total (including both endpoints), so $b=u_{k+2}$, not $u_{k+1}$.
• Sign errors with a negative common difference. A decreasing sequence still uses the same formulas; substitute the negative value of $d$ carefully and keep track of signs through every multiplication.
• Assuming every sequence with a sum formula is arithmetic. As the worked example in Section 5 shows, you must actually verify $u_n$ is linear in $n$ (or that consecutive differences are constant) before claiming the sequence is arithmetic.

Summary

• An arithmetic sequence has constant common difference $d=u_{n+1}-u_n$, defined recursively by $u_1$ and $u_{n+1}=u_n+d$.
• General term: $u_n = u_1+(n-1)d$, provable by induction from the recursive definition.
• Given two terms $u_m, u_n$, subtracting their general-term equations gives $d=\dfrac{u_n-u_m}{n-m}$ directly.
• The arithmetic mean of $a,b$ is $\dfrac{a+b}{2}$; inserting $k$ means between $a$ and $b$ uses $d=\dfrac{b-a}{k+1}$ over $k+2$ total terms.
• Sum of the first $n$ terms: $S_n=\dfrac{n}{2}(u_1+u_n) = \dfrac{n}{2}\big(2u_1+(n-1)d\big)$, derived by pairing the series with its reverse.
• Any term can be recovered from sums via $u_n=S_n-S_{n-1}$ for $n\geq2$, and $u_1=S_1$ — true for any sequence, not only arithmetic ones.
• Arithmetic sequences model constant additive change per step: seating arrangements, simple interest, fixed salary increases, and similar applied contexts.