A sequence is an ordered list of numbers. Each number in the sequence is called a term.

Notation: $a_1, a_2, a_3, a_4, \ldots, a_n, \ldots$

• $a_1$ is the first term
• $a_n$ is the $n$th term
• $n$ is the term number (position)

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

Common difference: $d = a_{n+1} - a_n$

Formulas

Explicit (Direct) Formula: $a_n = a_1 + (n - 1)d$

where:

• $a_n$ = $n$th term
• $a_1$ = first term
• $d$ = common difference
• $n$ = term number

Recursive Formula: $a_n = a_{n-1} + d, \quad a_1 = \text{(given)}$

Example: $3, 7, 11, 15, 19, \ldots$

• First term: $a_1 = 3$
• Common difference: $d = 4$
• Explicit: $a_n = 3 + (n-1)(4) = 3 + 4n - 4 = 4n - 1$
• Recursive: $a_n = a_{n-1} + 4, \quad a_1 = 3$
• 10th term: $a_{10} = 4(10) - 1 = 39$

Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms.

Common ratio: $r = \frac{a_{n+1}}{a_n}$

Formulas

Explicit (Direct) Formula: $a_n = a_1 \cdot r^{n-1}$

where:

• $a_n$ = $n$th term
• $a_1$ = first term
• $r$ = common ratio
• $n$ = term number

Recursive Formula: $a_n = a_{n-1} \cdot r, \quad a_1 = \text{(given)}$

Example: $2, 6, 18, 54, 162, \ldots$

• First term: $a_1 = 2$
• Common ratio: $r = 3$
• Explicit: $a_n = 2 \cdot 3^{n-1}$
• Recursive: $a_n = 3a_{n-1}, \quad a_1 = 2$
• 6th term: $a_6 = 2 \cdot 3^5 = 2 \cdot 243 = 486$

Identifying Sequence Type

Arithmetic: Check if differences are constant

• $7, 11, 15, 19, \ldots$ → differences: $4, 4, 4$ ✓

Geometric: Check if ratios are constant

• $3, 12, 48, 192, \ldots$ → ratios: $4, 4, 4$ ✓

Neither: If differences and ratios both vary

• $1, 4, 9, 16, \ldots$ (perfect squares) → neither

Sum of Arithmetic Sequence (Finite)

Sum of first $n$ terms: $S_n = \frac{n(a_1 + a_n)}{2}$

or

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

Example

Sum of first 10 terms of $3, 7, 11, 15, \ldots$: $S_{10} = \frac{10(3 + 39)}{2} = \frac{10(42)}{2} = 210$

Sum of Geometric Sequence (Finite)

Sum of first $n$ terms: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1$

Example

Sum of first 5 terms of $2, 6, 18, 54, \ldots$: $S_5 = 2 \cdot \frac{1 - 3^5}{1 - 3} = 2 \cdot \frac{1 - 243}{-2} = 2 \cdot \frac{-242}{-2} = 242$

Applications

• Arithmetic: Linear growth, evenly spaced values

  • Saving $50 per month
  • Theater seating rows

• Geometric: Exponential growth/decay

  • Population growth
  • Radioactive decay
  • Compound interest

a) Write the first five terms. b) Find the 20th term. c) Find the sum of the first 20 terms.

a) Find the 6th term. b) Find the sum of the first 8 terms.

For the 3rd term: $a_3 = a_1 + 2d = 14$

For the 7th term: $a_7 = a_1 + 6d = 30$

Solve the system by elimination:

Subtract first equation from second: $(a_1 + 6d) - (a_1 + 2d) = 30 - 14$ $4d = 16$ $d = 4$

Find $a_1$: Substitute $d = 4$ into first equation: $a_1 + 2(4) = 14$ $a_1 + 8 = 14$ $a_1 = 6$

Write the explicit formula: $a_n = a_1 + (n-1)d$ $a_n = 6 + (n-1)(4)$ $a_n = 6 + 4n - 4$ $a_n = 4n + 2$

Verify:

• $a_3 = 4(3) + 2 = 14$ ✓
• $a_7 = 4(7) + 2 = 30$ ✓

Answers:

• First term: $a_1 = 6$
• Common difference: $d = 4$
• Explicit formula: $a_n = 4n + 2$