Sequences Lecture Review

Arithmetic Sequences

Definition: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference, denoted by d.
General Form: If a1 is the first term, an arithmetic sequence can be written as:a1, a1+d, a1+2d, a_1+3d, \dots
n^{th} Term Formula: The formula to find any term (an) in an arithmetic sequence is:an = a1 + (n-1)d * Where: * an is the n^{th} term.
* a_1 is the first term.
* n is the term number.
* d is the common difference.

Informal Method (Gauss's Pairing): For sequences with a common difference of d=1, the sum can be found by pairing the first and last terms, the second and second-to-last terms, and so on. Each pair sums to the same value.
- Example 1: Sum of integers from 1 to 100
  - Sequence: 1, 2, 3, \dots, 99, 100
  - Number of terms (n): 100
  - Number of pairs: 100/2 = 50
  - Sum of each pair (a1+an): 1+100 = 101
  - Total Sum (S_n): 50 \times 101 = 5050
- Example 2: Sum of integers from 7 to 50
  - Sequence: 7, 8, 9, \dots, 50 (d=1)
  - First term (a_1): 7
  - Last term (a_n): 50
  - Number of terms (n): 50 - 7 + 1 = 44
  - Number of pairs: 44/2 = 22
  - Sum of each pair: 7+50 = 57
  - Total Sum (S_n): 22 \times 57 = 1254
General Formula for the Sum of an Arithmetic Sequence: The sum of the first n terms of an arithmetic sequence (Sn) is given by: Sn = \frac{n}{2}(a1 + an)
- Where:
  - n is the number of terms.
  - a_1 is the first term.
  - a_n is the last term.
- This formula works for all arithmetic sequences, regardless of the common difference.
- Note: Both the pairing method and the general sum formula are effective when the common difference d=1.

Example 1: Sum from 1 to 103 (d=1)
- Sequence: 1, 2, \dots, 103
- a1 = 1, an = 103, n = 103
- S_{103} = \frac{103}{2}(1 + 103) = \frac{103}{2}(104) = 103 \times 52 = 5356
Example 2: Sum from 10 to 103 (d=1)
- Sequence: 10, 11, \dots, 103
- a1 = 10, an = 103
- Number of terms (n): 103 - 10 + 1 = 94
- S_{94} = \frac{94}{2}(10 + 103) = 47 \times 113 = 5311
Example 3: Sum from 67 to 201 (d=1)
- Sequence: 67, 68, \dots, 201
- a1 = 67, an = 201
- Number of terms (n): 201 - 67 + 1 = 135
- S_{135} = \frac{135}{2}(67 + 201) = \frac{135}{2}(268) = 135 \times 134 = 18090
Example 4: Sum from 1 to 126 (d=1)
- Sequence: 1, 2, \dots, 126
- a1 = 1, an = 126, n = 126
- S_{126} = \frac{126}{2}(1 + 126) = 63 \times 127 = 7980
Example 5: Sum from 7 to 126 (d=1)
- Sequence: 7, 8, \dots, 126
- a1 = 7, an = 126
- Number of terms (n): 126 - 7 + 1 = 120
- S_{120} = \frac{120}{2}(7 + 126) = 60 \times 133 = 7980

Example 1: Sequence 4, 6, 8, \dots, 100 (Common Difference d=2)
- a1 = 4, an = 100, d = 2
- Step 1: Find the number of terms (n) using an = a1 + (n-1)d:
  - 100 = 4 + (n-1)2
  - 96 = (n-1)2
  - 48 = n-1
  - n = 49
- Step 2: Calculate the sum (S{49}) using Sn = \frac{n}{2}(a1 + an):
  - S_{49} = \frac{49}{2}(4 + 100) = \frac{49}{2}(104) = 49 \times 52 = 2548
Example 2: Sequence starting at 4, ending at 100, with d=3
- a1 = 4, an = 100, d = 3
- Step 1: Find the number of terms (n):
  - 100 = 4 + (n-1)3
  - 96 = (n-1)3
  - 32 = n-1
  - n = 33
- Step 2: Calculate the sum (S_{33}) using the formula:
  - S_{33} = \frac{33}{2}(4 + 100) = \frac{33}{2}(104) = 33 \times 52 = 1716
Example 3: Finding a specific term
- For an arithmetic sequence with a_1 = 5 and d=3:
  - The 66^{th} term (a{66}) is calculated as: a{66} = 5 + (66-1)3 = 5 + 65 \times 3 = 5 + 195 = 200

Definition: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by r.
Example: Sum of terms in a decreasing geometric sequence
- The sequence 64, 16, 4, 1 is a geometric sequence.
- First term (a_1): 64
- Common ratio (r): 16/64 = 1/4
- Sum of these terms: 64 + 16 + 4 + 1 = 85

Order of Operations (PEMDAS/BODMAS): Follow the standard order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
- Example: 7^2 - 49
  - Exponents first: 49 - 49 = 0
Powers and Basic Arithmetic:
- 53 \times 53 = 2809
- 2809 - 1173 - 5 = 1631
Sum of Powers of 10:
- 1000 + 100 + 10 + 1 = 1111
Gematria: A system of assigning numerical values to a word or name (mentioned in the context but not further explained or applied).