Sequences and Arithmetic Progressions

Get­ting started#

Think of a sequence as num­bers stand­ing in a queue, in a fixed order. Add them up and you have a series. We will first see two ways of describ­ing a sequence, by a for­mula or by a rule that builds each term from the ones before it. Then we spend most of our time on the friend­liest kind, the arith­metic pro­gres­sion, where you keep adding the same amount: its gen­eral term, the sum of $n$ terms, and the arith­metic mean.

Sequences and series#

A sequence is a func­tion on $\mathbf{N}$: $a_1, a_2, a_3, \ldots$, with $a_n$ the gen­eral term. You can describe it by a for­mula, so $a_n = n^2 - 1$ gives $0, 3, 8, 15, \ldots$, or recur­sively, so $a_1 = 2$, $a_{n+1} = 3a_n - 1$ gives $2, 5, 14, 41, \ldots$ The famous Fibonacci sequence is recur­sive too: $a_1 = a_2 = 1$, $a_{n+2} = a_{n+1} + a_n$.

A series is just the sum $\displaystyle a_1 + a_2 + \cdots + a_n = \sum_{k=1}^{n} a_k$.

Arith­metic pro­gres­sions#

In an AP you move by a con­stant com­mon dif­fer­ence $d$: $a, a + d, a + 2d, \ldots$

$$\displaystyle a_n = a + (n - 1)d, \qquad S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l),$$

where $l$ is the last term. Where does the sum for­mula come from? Write $S_n$ for­wards, then back­wards under­neath, and add the two lines. Every pair adds up to the same thing, $a + l$, and that is the whole trick.

A handy fact: if you add, sub­tract, mul­ti­ply or divide every term by the same con­stant, you still have an AP.

Arith­metic mean#

The AM of $a$ and $b$ is $\displaystyle \tfrac{a + b}{2}$, the num­ber sit­ting exactly halfway, so that $a, A, b$ is an AP. If you want to insert $n$ num­bers between $a$ and $b$ mak­ing an AP, take $\displaystyle d = \tfrac{b - a}{n + 1}$.

Try these your­self#

1. Write down the first four terms of each: $\displaystyle a_n = \tfrac{2n - 1}{n + 1}$; $a_n = (-1)^n 3^n$; $a_1 = 4$, $\displaystyle a_{n+1} = \tfrac{a_n}{2} + 1$.
2. Find the $25$th term and the sum of $25$ terms of $3, 8, 13, \ldots$
3. Which term of $\sqrt{3}, 3\sqrt{3}, 5\sqrt{3}, \ldots$ is $39\sqrt{3}$?
4. The $5$th term of an AP is $19$ and the sum of the first $10$ terms is $185$. Find $a$ and $d$.
5. Find the sum of all odd num­bers between $100$ and $300$.
6. The sum of $n$ terms of an AP is $3n^2 + 2n$. Find its $n$th term.
7. If the $p$th term is $q$ and the $q$th term is $p$ ($p \ne q$), show that the $(p + q)$th term is $0$.
8. Insert $6$ arith­metic means between $-5$ and $30$.
9. A man saves ₹$800$ in the first month and increases his sav­ing by ₹$150$ each month. How much has he saved in $2$ years?
10. The angles of a hexa­gon are in AP with the small­est $95^\circ$. Find the com­mon dif­fer­ence.

Answers to check against#