Chapter 24 — Sequences and Series
Table of Contents
24.1 Arithmetic Sequences
Definition. An arithmetic sequence (or arithmetic progression, AP) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference $d$.
\[a,\; a+d,\; a+2d,\; a+3d,\; \ldots\]The $n$-th term is:
\[a_n = a + (n-1)d\]where $a = a_1$ is the first term.
Example 24-1. The sequence $3, 7, 11, 15, 19, \ldots$ has first term $a = 3$ and common difference $d = 4$.
The 20th term is $a_{20} = 3 + (19)(4) = 79$.
Example 24-2. If the 5th term of an AP is $13$ and the 12th term is $34$, find the first term and common difference.
$a + 4d = 13$ and $a + 11d = 34$. Subtracting: $7d = 21$, so $d = 3$ and $a = 1$.
Exercise 24-1. Find the common difference and the next three terms: $3, 7, 11, 15, \ldots$
Exercise 24-2. What is the 20th term of $3, 7, 11, 15, \ldots$?
Exercise 24-3. The first term of an AP is $7$ and the common difference is $-3$. What is the 30th term?
24.2 Arithmetic Series
Definition. An arithmetic series is the sum of the terms of an arithmetic sequence.
\[S_n = a + (a+d) + (a+2d) + \cdots + (a+(n-1)d)\]Formula:
\(\boxed{S_n = \frac{n}{2}\bigl(2a + (n-1)d\bigr) = \frac{n(a_1 + a_n)}{2}}\)
Gauss’s Trick. Write the sum forwards and backwards and add:
\(S = 1 + 2 + 3 + \cdots + n\) \(S = n + (n-1) + \cdots + 1\) \(2S = (n+1) + (n+1) + \cdots + (n+1) = n(n+1)\) \(S = \frac{n(n+1)}{2}\)
Example 24-3. Find the sum $3 + 7 + 11 + \cdots + 79$.
Here $a = 3$, $d = 4$, $a_n = 79$. From $a_n = a + (n-1)d$: $79 = 3 + 4(n-1)$, so $n = 20$.
\(S = \frac{20(3 + 79)}{2} = \frac{20 \cdot 82}{2} = 820.\)
Exercise 24-4. Evaluate $3 + 7 + 11 + 15 + \cdots + 99$.
Exercise 24-5. Evaluate $42 + 35 + 28 + 21 + \cdots + (-77)$.
24.3 Geometric Sequences
Definition. A geometric sequence (or geometric progression, GP) is a sequence where each term is obtained by multiplying the previous term by a fixed constant called the common ratio $r$.
\[a,\; ar,\; ar^2,\; ar^3,\; \ldots\]The $n$-th term is:
\(a_n = a \cdot r^{n-1}\)
Example 24-4. The sequence $2, 6, 18, 54, \ldots$ has $a = 2$ and $r = 3$. The 8th term is $2 \cdot 3^7 = 4374$.
Exercise 24-6. Find the common ratio and 10th term of $5, 10, 20, 40, \ldots$
Exercise 24-7. The 3rd term of a GP is $12$ and the 6th term is $96$. Find the first term and common ratio.
24.4 Geometric Series
Definition. A geometric series is the sum of a geometric sequence.
\[S_n = a + ar + ar^2 + \cdots + ar^{n-1}\]Formula (for $r \neq 1$):
\(\boxed{S_n = \frac{a(1 - r^n)}{1 - r} = \frac{a(r^n - 1)}{r - 1}}\)
Derivation. Let $S = a + ar + ar^2 + \cdots + ar^{n-1}$. Then:
\[rS = ar + ar^2 + \cdots + ar^n\]Subtracting: $S - rS = a - ar^n$, hence $S(1 - r) = a(1 - r^n)$.
Example 24-5. Find $2 + 6 + 18 + 54 + \cdots$ (8 terms).
$a = 2$, $r = 3$, $n = 8$:
\(S_8 = \frac{2(3^8 - 1)}{3 - 1} = \frac{2(6561 - 1)}{2} = 6560.\)
Exercise 24-8. Find the sum of the first 12 terms of $5, 10, 20, 40, \ldots$
24.5 Infinite Series
| Infinite Geometric Series. If $ | r | < 1$, the infinite geometric series converges: |
If $|r| \geq 1$, the series diverges (the sum does not exist as a finite number).
Example 24-6 (Zeno’s Paradox). To reach a wall, you must first go half the distance, then half again, etc.:
\[\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \frac{1/2}{1 - 1/2} = 1.\]So the total distance is exactly $1$ — you do reach the wall!
Example 24-7. Express $0.\overline{36} = 0.363636\ldots$ as a fraction.
\(0.\overline{36} = \frac{36}{100} + \frac{36}{10000} + \cdots = \frac{36/100}{1 - 1/100} = \frac{36}{99} = \frac{4}{11}.\)
Convergence and Divergence.
- A series converges if the sequence of partial sums approaches a finite value.
- A series diverges if the partial sums grow without bound or oscillate.
- Necessary condition: If $\sum a_n$ converges, then $a_n \to 0$. But the converse is false! (The harmonic series $\sum 1/n$ diverges despite $1/n \to 0$.)
Comparison Test (informal). If $0 \leq a_n \leq b_n$ for all $n$, and $\sum b_n$ converges, then $\sum a_n$ converges too. If $\sum a_n$ diverges, then $\sum b_n$ diverges too.
Exercise 24-9. Convert $0.\overline{142857}$ to a fraction.
Exercise 24-10. Does $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2}$ converge? (Hint: compare with $\frac{1}{n(n-1)}$ for $n \geq 2$.)
24.6 Notation, Recursion, and Means
Sigma and Pi Notation
Sigma notation (summation):
\[\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n\]Pi notation (product):
\[\prod_{k=1}^{n} a_k = a_1 \cdot a_2 \cdots a_n\]In particular, $n! = \prod_{k=1}^{n} k$.
Sequences via Explicit and Recursive Definitions
- Explicit formula: $a_n = f(n)$. Example: $a_n = 2n + 1$.
- Recursive formula: defines $a_n$ in terms of previous terms plus initial conditions. Example: $a_1 = 1$, $a_n = 2a_{n-1} + 1$.
The Fibonacci Sequence. $F_1 = 1$, $F_2 = 1$, $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$.
\[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots\]This is one of the most famous recursive sequences in mathematics.
Arithmetic and Geometric Means
Arithmetic Mean (AM) of $a$ and $b$: $\dfrac{a+b}{2}$.
Geometric Mean (GM) of positive $a$ and $b$: $\sqrt{ab}$.
The AM is always $\geq$ the GM (by AM-GM inequality from Chapter 22).
In an arithmetic sequence, each term is the AM of its neighbors: $a_n = \dfrac{a_{n-1} + a_{n+1}}{2}$.
In a geometric sequence (positive terms), each term is the GM of its neighbors: $a_n = \sqrt{a_{n-1} \cdot a_{n+1}}$.
Quick-Reference Formula Table
| Quantity | Arithmetic | Geometric |
|---|---|---|
| $n$-th term | $a + (n-1)d$ | $ar^{n-1}$ |
| Finite sum ($n$ terms) | $\dfrac{n}{2}(2a + (n-1)d)$ | $\dfrac{a(1 - r^n)}{1 - r}$ |
| Infinite sum | diverges (unless $d = 0$) | $\dfrac{a}{1 - r}$ if $\lvert r\rvert < 1$ |
| Mean of two values | $\dfrac{a + b}{2}$ | $\sqrt{ab}$ |
Problems to Solve for Chapter 24
Problem 434. In AP $22, 16, 10, \ldots$, what is the first negative term? (AHSME 1955)
Problem 435. Find $1 + 3 + 5 + \cdots + (2n - 1)$.
Problem 436. The sum of 40 consecutive positive odd integers is $2{,}000$. Find the largest. (MAΘ 1987)
Problem 437. Find $\displaystyle\sum_{k=1}^{200} (3k + 5)$.
Problem 438. The sum of the first $n$ terms of a sequence is $n^2 + n$. Find $a_{10}$ and a formula for $a_n$. (AHSME 1973)
Problem 439. Express $0.4\overline{23}$ as a fraction.
Problem 440. The harmonic mean of $a$ and $b$ is $\dfrac{2ab}{a+b}$. Find the harmonic mean of $1$ and $\frac{1}{3}$ by finding $\frac{2}{1 + 3}$.
Problem 441. What rational number equals $2.71828\overline{1828}$?
Problem 442. Show that any 4 consecutive Fibonacci numbers $F_n, F_{n+1}, F_{n+2}, F_{n+3}$ satisfy $F_{n+3}^2 - F_n^2 = F_{n+1} F_{n+2} + F_{n+2} F_{n+3}$.
Problem 443. A ball is dropped from 100 feet. Each time it bounces, it rises to $\frac{2}{3}$ of its previous height. What is the total distance traveled (up and down combined)? (AHSME 1961)
Problem 444. Find the sum $\dfrac{2}{1 \cdot 3} + \dfrac{2}{3 \cdot 5} + \dfrac{2}{5 \cdot 7} + \cdots + \dfrac{2}{(2n-1)(2n+1)}$. (Hint: partial fractions.)
Problem 445. Find the sum $1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots$ (MATHCOUNTS 1986)
Problem 446. Evaluate $1 + 2 + 3 + \cdots + n$ for $n = 100$. (Classic)
Problem 447. Find the 10th term of the sequence $1, 1, 2, 3, 5, 8, \ldots$ (MATHCOUNTS 1984)
Problem 448. The first three terms of an AP are $x - 1$, $x + 1$, $2x + 3$. Find $x$. (MATHCOUNTS 1984)
Problem 449. The sum of an infinite GP is 4 and the common ratio is $1/4$. Find the first term.
Problem 450. Three zeros are inserted between every pair of digits of $1, 2, 3, \ldots, 9$ to form $1000200030004\ldots$. This is then read as a single number. If we call it $N$, find $N$.
Problem 451. The first term of a GP is $a$, fourth is $b$, tenth is $c$. Prove $b^2 = a \cdot c$ is false, but $b^3 = a^2 c$ is false too — find the correct relationship.
Problem 452. Find $\displaystyle\sum_{k=0}^{n} r^k$ for $r \neq 1$.
Problem 453. Evaluate $\displaystyle\sum_{k=1}^{100} \frac{1}{k(k+1)}$.
Problem 454. The measures of the angles of a convex pentagon form an AP. Find the measure of the 3rd (middle) angle. (MAΘ 1990)
Problem 455. The interior angles of a polygon form an AP with common difference $25°$. Find all possible numbers of sides.
Problem 456. The sequence $a_1, a_2, \ldots$ satisfies $a_1 = 19$, $a_9 = 99$, and for all $n \geq 3$, $a_n$ is the arithmetic mean of the first $n-1$ terms. Find $a_2$. (AHSME 1992)