Overview
This lecture covers the differences between sequences and series, how to identify and work with arithmetic and geometric sequences, formulas for finding specific terms and sums, and the use of summation notation.
Sequences vs. Series
- A sequence is a list of numbers separated by commas.
- A series is the sum of terms in a sequence.
Sequence Notation & Terms
- The nth term is denoted as ( a_n ); subscript indicates the term's position.
- ( n ) refers to the position; ( a_n ) to the value at that position.
Arithmetic Sequences
- Arithmetic sequences add the same value, called the common difference (( d )), each time.
- Formula: ( a_n = a_1 + d(n-1) ).
- Recursive formula: ( a_1 = \text{first term},\ a_n = a_{n-1} + d ).
- Example: For ( a_1 = 3, d = 2 ), the 6th term is ( 3 + 2(6-1) = 13 ).
- To find the sum of the first ( n ) terms: ( S_n = \frac{n}{2}(a_1 + a_n) ).
Geometric Sequences
- Geometric sequences multiply by the same value, the common ratio (( r )), each time.
- Formula: ( a_n = a_1 \cdot r^{n-1} ).
- Recursive formula: ( a_1 = \text{first term},\ a_n = a_{n-1} \cdot r ).
- Example: For ( a_1 = 5, r = 3 ), 5th term is ( 5 \cdot 3^{4} ).
- Sum of first ( n ) terms: ( S_n = a_1 \frac{1-r^n}{1-r} ) (for ( r \neq 1 )).
Infinite Geometric Series
- An infinite series converges only if ( -1 < r < 1 ).
- Formula: ( S = \frac{a_1}{1-r} ).
- Example: For ( a_1 = 100, r = 1/2 ), sum is ( \frac{100}{1-1/2} = 200 ).
Writing Formulas Given Terms
- For arithmetic sequences, set up two equations with the formula to solve for ( a_1 ) and ( d ).
- For geometric sequences, use two known terms to solve for ( a_1 ) and ( r ) via substitution.
Summation Notation (Sigma)
- Sigma notation ( \sum ) means sum; lower index shows starting value, upper shows ending.
- Substitute each integer into the formula and add terms sequentially.
Key Terms & Definitions
- Sequence — An ordered list of numbers.
- Series — The sum of the terms in a sequence.
- Common difference (( d )) — The amount added in arithmetic sequences.
- Common ratio (( r )) — The multiplier in geometric sequences.
- Recursive formula — Defines each term based on the previous term.
- Sigma notation (( \sum )) — Symbol representing summation of terms.
Action Items / Next Steps
- Practice identifying and writing formulas for arithmetic and geometric sequences.
- Complete exercises using summation notation for both types of sequences.
- Review sequence and series formulas for upcoming quizzes or exams.