• $a_1$ 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

• First term: $a_1 = 5$
• Common difference: $d = 9 - 5 = 4$

Explicit formula: $a_n = a_1 + (n-1)d$ $a_n = 5 + (n-1)(4)$ $a_n = 5 + 4n - 4$ $a_n = 4n + 1$

Recursive formula: $a_n = a_{n-1} + 4, \quad a_1 = 5$

Find the 20th term: $a_{20} = 4(20) + 1 = 80 + 1 = 81$

Verify: We can check by adding $d$ nineteen times to $a_1$: $5 + 19(4) = 5 + 76 = 81$ ✓

Answers:

• Explicit: $a_n = 4n + 1$
• Recursive: $a_n = a_{n-1} + 4, a_1 = 5$
• 20th term: $81$

$a_1 = 5$ $a_2 = 5 + 3 = 8$ $a_3 = 8 + 3 = 11$ $a_4 = 11 + 3 = 14$ $a_5 = 14 + 3 = 17$

First five terms: 5, 8, 11, 14, 17

Part (b): $a_{20} = a_1 + (20-1)d = 5 + 19(3) = 5 + 57 = 62$

Part (c): Sum formula: $S_n = \frac{n(a_1 + a_n)}{2}$

$S_{20} = \frac{20(5 + 62)}{2} = \frac{20(67)}{2} = 10(67) = 670$

Find the sum of first 8 terms: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ $S_8 = 3 \cdot \frac{1 - 2^8}{1 - 2}$ $S_8 = 3 \cdot \frac{1 - 256}{-1}$ $S_8 = 3 \cdot \frac{-255}{-1}$ $S_8 = 3 \cdot 255 = 765$

Verify the sequence: $3, 6, 12, 24, 48, 96, 192, 384$ Sum: $3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 = 765$ ✓

Answers:

• 8th term: $384$
• Sum of first 8 terms: $765$

$a_6 = 2 \cdot 3^{6-1} = 2 \cdot 3^5 = 2 \cdot 243 = 486$

Part (b): Sum formula: $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$ (for $r \neq 1$)

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

$= 2 \cdot \frac{6561 - 1}{2}$

$= 2 \cdot \frac{6560}{2}$

$= 2 \cdot 3280$

$= 6560$