Sequences & Series

📊 Sequences & Series — Arithmetic Sequences

Part 1 of 7

What Is a Sequence?

A sequence is an ordered list of numbers following a pattern. Each number is called a term.

$a_1, a_2, a_3, \ldots, a_n, \ldots$

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms:

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

where $d = a_{n+1} - a_n$ is the common difference.

Examples

• $2, 5, 8, 11, 14, \ldots$ → $d = 3$
• $20, 15, 10, 5, 0, \ldots$ → $d = -5$
• $\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots$ → $d = \frac{1}{2}$

📝 Finding Terms and Sums

The $n$th Term Formula

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

Example: $a_1=3, d=4$. Find $a_{20}$:

$a_{20} = 3 + 19(4) = 79$

Arithmetic Series (Sum)

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

Example: Sum the first $100$ positive integers.

$a_1 = 1, a_{100} = 100, n = 100$

$S_{100} = \frac{100}{2}(1+100) = 50 \cdot 101 = 5050$

💡 This is Gauss's famous result! He supposedly computed this as a schoolboy.

🧩 Applications

Finding Unknown Terms

If $a_5 = 17$ and $a_{12} = 45$:

$a_{12} - a_5 = (12-5)d$

$45 - 17 = 7d \implies d = 4$

$a_1 = a_5 - 4d = 17 - 16 = 1$

Arithmetic Mean

The arithmetic mean of $a$ and $b$ is $\frac{a+b}{2}$, which is the term between them in an arithmetic sequence.

Insert 3 arithmetic means between 2 and 14:

$a_1 = 2, a_5 = 14$. $14 = 2+4d \implies d = 3$.

Sequence: $2, 5, 8, 11, 14$.

Arithmetic Sequences Quiz 🎯

Arithmetic Sequence Practice 🧮

1) $a_1=5, d=7$. Find $a_{10}$:

2) Sum of first $50$ positive integers: $S_{50}$ = ?

3) $a_3=11, a_7=23$. Find $d$:

Arithmetic Sequences Properties 🔽

Exit Quiz ✅

📊 Geometric Sequences

Part 2 of 7

Definition

A geometric sequence has a constant ratio between consecutive terms:

$a_n = a_1 \cdot r^{n-1}$

where $r = \frac{a_{n+1}}{a_n}$ is the common ratio.

Examples

💡 If $|r| > 1$, terms grow; if $|r| < 1$, terms shrink; if $r < 0$, terms alternate sign.

📝 Finite Geometric Series

$S_n = a_1 \cdot \frac{1-r^n}{1-r} \quad (r \neq 1)$

Example: Find $S_8$ for $a_1=3, r=2$

$S_8 = 3 \cdot \frac{1-2^8}{1-2} = 3 \cdot \frac{1-256}{-1} = 3 \cdot 255 = 765$

Geometric Mean

The geometric mean of positive $a$ and $b$ is $\sqrt{ab}$.

Insert a geometric mean between 4 and 16: $\sqrt{4 \cdot 16} = \sqrt{64} = 8$

Sequence: $4, 8, 16$ with $r = 2$.

📊 Arithmetic vs. Geometric

Key Insight

Arithmetic sequences grow by addition → linear growth.

Geometric sequences grow by multiplication → exponential growth (or decay if $|r|<1$).

Geometric Sequences Quiz 🎯

Geometric Sequence Practice 🧮

1) $a_1=5, r=2$. Find $a_8$:

2) $a_1=1000, r=\frac{1}{2}$. Find $a_5$:

3) $S_4$ for $a_1=3, r=4$: (use the sum formula)

Geometric Sequences Properties 🔽

Exit Quiz ✅

♾️ Infinite Geometric Series

Part 3 of 7

When Does an Infinite Sum Converge?

For a geometric series with $|r| < 1$, the partial sums approach a finite limit:

$S_\infty = \frac{a_1}{1-r} \quad \text{when } |r| < 1$

If $|r| \geq 1$, the series diverges (no finite sum).

Why It Works

As $n \to \infty$ and $|r| < 1$, $r^n \to 0$:

$S_n = a_1 \cdot \frac{1-r^n}{1-r} \to \frac{a_1}{1-r}$

Example

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \frac{1}{1-\frac{1}{2}} = 2$

📝 Worked Examples

Example 1: $\sum_{n=0}^{\infty} 3\left(\frac{2}{5}\right)^n$

$a_1 = 3, r = \frac{2}{5}$. Since $|r| < 1$:

$S = \frac{3}{1-\frac{2}{5}} = \frac{3}{\frac{3}{5}} = 5$

Example 2: Repeating Decimal $0.\overline{36}$

$0.363636\ldots = 0.36 + 0.0036 + 0.000036 + \cdots$

$a_1 = 0.36, r = 0.01$

$S = \frac{0.36}{1-0.01} = \frac{0.36}{0.99} = \frac{36}{99} = \frac{4}{11}$

Example 3: Does $\sum 5(1.1)^n$ converge?

$r = 1.1 > 1$ → Diverges! No finite sum.

🌍 Applications

Bouncing Ball Total Distance

Ball dropped from $h = 10$ m, rebounds to $\frac{3}{5}$ of height each time.

Total distance = down + up + down + up + ...

$= 10 + 2 \sum_{n=1}^{\infty} 10\left(\frac{3}{5}\right)^n = 10 + 2 \cdot \frac{10 \cdot \frac{3}{5}}{1-\frac{3}{5}} = 10 + 2 \cdot \frac{6}{\frac{2}{5}} = 10 + 30 = 40$ m

Drug Dosage (Pharmacokinetics)

If 60% of a drug remains after each dose period and dose is 200 mg:

Long-term level = $\frac{200}{1-0.6} = 500$ mg (steady state)

Infinite Series Quiz 🎯

Infinite Series Calculations 🧮

1) $\sum_{n=0}^{\infty} 8\left(\frac{1}{4}\right)^n$ = ?

2) $0.\overline{7} = \frac{?}{9}$. Enter the numerator.

3) $\sum_{n=1}^{\infty} \frac{6}{10^n}$ = ? (Enter as a fraction like "2/3")

Convergence Concepts 🔽

Exit Quiz ✅

🔢 Sigma Notation & Series

Part 4 of 7

Sigma (Summation) Notation

$\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n$

• $k$ = index of summation (dummy variable)
• Lower limit: starting value
• Upper limit: ending value

Examples

📐 Properties of Summation

Linearity Rules

$\sum_{k=1}^n (a_k + b_k) = \sum_{k=1}^n a_k + \sum_{k=1}^n b_k$

$\sum_{k=1}^n c \cdot a_k = c \cdot \sum_{k=1}^n a_k$

Useful Closed-Form Sums

Telescoping Sums

$\sum_{k=1}^n [f(k)-f(k-1)] = f(n)-f(0)$ — most terms cancel!

📝 Writing in Sigma Notation

Example 1: $2+4+6+8+\cdots+100$

Pattern: $a_k = 2k$, from $k=1$ to $50$.

$\sum_{k=1}^{50} 2k = 2\sum_{k=1}^{50} k = 2 \cdot \frac{50(51)}{2} = 2550$

Example 2: $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$

Pattern: $a_k = (-\frac{1}{2})^k$, from $k=0$ to $\infty$.

$\sum_{k=0}^{\infty}\left(-\frac{1}{2}\right)^k = \frac{1}{1+\frac{1}{2}} = \frac{2}{3}$

Example 3: $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots+\frac{1}{99\cdot 100}$

Partial fractions: $\frac{1}{k(k+1)} = \frac{1}{k}-\frac{1}{k+1}$ → telescoping!

$\sum_{k=1}^{99}\left(\frac{1}{k}-\frac{1}{k+1}\right) = 1 - \frac{1}{100} = \frac{99}{100}$

Sigma Notation Quiz 🎯

Sigma Calculations 🧮

1) $\sum_{k=1}^{4} k^2$ = ?

2) $\sum_{k=1}^{10} k$ = ?

3) $\sum_{k=1}^{6} 5$ = ?

Sigma Concepts 🔽

Exit Quiz ✅

🔁 Recursive Sequences & Special Sequences

Part 5 of 7

Recursive vs. Explicit Formulas

The Fibonacci Sequence

$F_1=1, F_2=1, F_n = F_{n-1}+F_{n-2}$

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots$

Each term is the sum of the two preceding terms.

💡 The ratio of consecutive Fibonacci numbers approaches the Golden Ratio $\phi = \frac{1+\sqrt{5}}{2} \approx 1.618$.

📝 Working with Recursive Formulas

Example 1: $a_1=2, a_n=3a_{n-1}-1$

$a_1 = 2$ $a_2 = 3(2)-1 = 5$ $a_3 = 3(5)-1 = 14$ $a_4 = 3(14)-1 = 41$

Example 2: Converting Recursive → Explicit

Given: $a_1 = 5, a_n = a_{n-1}+4$

This is arithmetic with $d=4$: $a_n = 5+4(n-1) = 4n+1$.

Example 3: Logistic Growth

$P_{n+1} = r \cdot P_n(1-P_n)$

This recursive formula models population growth with limited resources. Unlike geometric growth, it accounts for carrying capacity.

🌟 Special Sequences

Triangular Numbers

$1, 3, 6, 10, 15, 21, \ldots$ → $T_n = \frac{n(n+1)}{2}$

Square Numbers

$1, 4, 9, 16, 25, \ldots$ → $S_n = n^2$

Factorial Sequence

$1, 1, 2, 6, 24, 120, \ldots$ → $n! = n(n-1)(n-2)\cdots 1$

Powers of 2

$1, 2, 4, 8, 16, 32, \ldots$ → $a_n = 2^{n-1}$

Harmonic Sequence

$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$ → $a_n = \frac{1}{n}$

The harmonic series $\sum \frac{1}{n}$ diverges — even though terms go to 0!

Recursive Sequences Quiz 🎯

Recursive Calculations 🧮

For $a_1=3, a_n=a_{n-1}^2-2$:

1) $a_2$ = ?

2) $a_3$ = ?

3) What is $5!$ (5 factorial)?

Special Sequences Concepts 🔽

Exit Quiz ✅

🧮 Binomial Theorem

Part 6 of 7

Pascal's Triangle

$\begin{array}{c} 1 \\ 1 \quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 3 \quad 1 \\ 1 \quad 4 \quad 6 \quad 4 \quad 1 \end{array}$

Each number is the sum of the two numbers above it.

Binomial Coefficients

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Read "$n$ choose $k$" — the number of ways to select $k$ items from $n$.

The Binomial Theorem

$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

📝 Expanding Binomials

Example 1: $(x+y)^4$

$\binom{4}{0}x^4 + \binom{4}{1}x^3y + \binom{4}{2}x^2y^2 + \binom{4}{3}xy^3 + \binom{4}{4}y^4$

$= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

Example 2: $(2x-3)^3$

Let $a=2x, b=-3$:

$(2x)^3 + 3(2x)^2(-3) + 3(2x)(-3)^2 + (-3)^3$

$= 8x^3 - 36x^2 + 54x - 27$

Finding a Specific Term

The $(k+1)$th term of $(a+b)^n$ is: $\binom{n}{k}a^{n-k}b^k$

Find the 4th term of $(x+2)^6$: $k=3$

$\binom{6}{3}x^3(2)^3 = 20 \cdot x^3 \cdot 8 = 160x^3$

🔑 Key Properties

Symmetry

$\binom{n}{k} = \binom{n}{n-k}$

Example: $\binom{7}{2} = \binom{7}{5} = 21$

Sum of Row

$\sum_{k=0}^n \binom{n}{k} = 2^n$

(Set $a=b=1$ in the binomial theorem)

Alternating Sum

$\sum_{k=0}^n (-1)^k\binom{n}{k} = 0$

(Set $a=1, b=-1$)

Pascal's Rule

$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$

This is why each entry in Pascal's triangle is the sum of the two above it!

Binomial Theorem Quiz 🎯

Binomial Calculations 🧮

1) $\binom{8}{3}$ = ?

2) The coefficient of $x^4$ in $(x+2)^5$ is: (include the power of 2)

3) The sum of row 6 of Pascal's triangle: $\sum_{k=0}^6 \binom{6}{k}$ = ?

Binomial Concepts 🔽

Exit Quiz ✅

🎯 Sequences & Series — Full Synthesis

Part 7 of 7

Master Summary

Key Decision Tree

Is there a common difference? → Arithmetic

Is there a common ratio? → Geometric

Is it defined by previous terms? → Recursive

Does it involve $\binom{n}{k}$? → Binomial expansion

🗺️ Problem-Solving Strategies

Finding the Pattern

1. Compute differences: $a_2-a_1, a_3-a_2, \ldots$ → constant? → arithmetic
2. Compute ratios: $a_2/a_1, a_3/a_2, \ldots$ → constant? → geometric
3. Check second differences → constant? → quadratic sequence

AP Exam Tips

• Know both sum formulas cold
• Practice converting between recursive and explicit
• For convergence questions: only geometric with $|r|<1$
• Telescoping sums: try partial fractions
• Binomial: know how to find a specific term without expanding everything

📝 Mixed Practice

Problem 1

Sequence: $2, 6, 18, 54, \ldots$

Ratios: $6/2=3$ → Geometric, $r=3$. $a_n = 2(3)^{n-1}$.

Problem 2

$\sum_{k=1}^{\infty} \frac{4}{5^k}$

$a_1 = 4/5, r = 1/5$. $S = \frac{4/5}{1-1/5} = \frac{4/5}{4/5} = 1$.

Problem 3

Find the coefficient of $x^2y^3$ in $(x+y)^5$:

$\binom{5}{3} = 10$.

Problem 4

Find the sum: $1 + 3 + 5 + \cdots + 99$

Arithmetic: $a_1=1, d=2, a_n=99$. $n = 50$. $S_{50} = \frac{50}{2}(1+99) = 2500$.

Synthesis Quiz 🎯

Mixed Calculations 🧮

1) Sum of first 20 terms of $5, 8, 11, 14, \ldots$:

2) $\sum_{n=1}^{\infty} 10(0.1)^n$ = ? (Enter as a fraction)

3) $\binom{7}{3}$ = ?

Master Classification 🔽

Final Exit Quiz ✅