Sequences, Series and Arithmetic Progressions: Quick Revision

A brief note on Sequences, Series and Arithmetic Progressions. Read it before a test. Try each question at the end before you open its answer.

Sequences and series

• A sequence is a function on $1, 2, 3, \dots$; its $n$th term is $a_n$.
• A formula gives $a_n$ at once. A recurrence builds each term from earlier ones.
• A series adds the terms: $S_n = a_1 + \dots + a_n = \sum_{k=1}^{n} a_k$.
• $a_1 = S_1$, and $a_n = S_n - S_{n-1}$ for $n \geq 2$.

Arithmetic progressions

• Each term is the one before plus $d$: $a_n = a + (n - 1)d$.
• From the end: the $n$th term from the end is $l - (n - 1)d$.
• $S_n = \dfrac{n}{2}\left(2a + (n - 1)d\right) = \dfrac{n}{2}(a + l)$.
• A sequence is an AP exactly when $a_n$ is linear in $n$, or $S_n$ is a quadratic in $n$ with no constant term.
• Adding, subtracting, multiplying or dividing every term by a constant keeps an AP.
• Choose three numbers as $a - d, a, a + d$, and four as $a - 3d, a - d, a + d, a + 3d$.

Arithmetic means

• The AM of $a$ and $b$ is $\dfrac{a + b}{2}$; $a$, $b$, $c$ are in an AP exactly when $2b = a + c$.
• $m$ means between $a$ and $b$: $d = \dfrac{b - a}{m + 1}$, and the $j$th mean is $a + jd$.
• The $m$ means add up to $m \times \dfrac{a + b}{2}$.

A worked example

An AP has $a_2 = 11$, and the sum of its first $9$ terms is $207$. Find its first term and common difference.

1. Use the sum. $S_9 = \dfrac{9}{2}(a_1 + a_9)$, and $a_1 + a_9 = 2a_5$, because they are the same distance from the middle. So $S_9 = 9a_5$.
2. The middle term. $9a_5 = 207$, so $a_5 = 23$.
3. The difference. $a_5 - a_2 = 3d = 12$, so $d = 4$.
4. The first term. $a = a_2 - d = 7$.

Practice with new numbers

A program to try

The program checks the formulas of this note on one AP: the $n$th term, the sum, recovering the terms from the sums, and the middle term as a mean.

Mistakes to avoid

• $a + nd$ for the $n$th term. It is $a + (n - 1)d$.
• $a_n = S_n - S_{n-1}$ at $n = 1$. Use $a_1 = S_1$.
• Dividing by $m$ for $m$ means. There are $m + 1$ steps.
• Rejecting a second root for $n$. Two values can both be right when some terms add up to zero.
• $b^2 = ac$ for an AP. That is the geometric test. An AP needs $2b = a + c$.

Check yourself

1. Find the $15$th term of the AP $-4, 1, 6, \dots$.

2. The partial sums of a sequence are $S_n = 5n^2$. Find $a_3$.

3. Find the sum $9 + 13 + 17 + \dots$ to $12$ terms.

4. Insert $2$ arithmetic means between $-5$ and $13$.

5. Four numbers in an AP add up to $40$. What do the two middle ones add up to?