Comprehensive Study Guide to Mathematical Sequences

Pattern Recognition and Sequence Identification

• Continuing the Sequence: Sequences are identified by observing the underlying pattern and applying it to determine subsequent values.
  • Sequence 1: $4, 1, 8, 12, ...$ (The pattern is adding 4 to each term).
  • Sequence 2: $1, 2, 4, 8, ...$ (The pattern is multiplying each term by 2).
  • Sequence 3: $1, 1, 2, 3, 5, ...$ (The pattern follows the Fibonacci rule where the sum of the previous two terms equals the next).
  • Sequence 4: $1, 4, 9, 16, 25, ...$ (The pattern consists of perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2, ...$).
  • Sequence 5: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...$ (The pattern follows unit fractions with increasing denominators).
  • Sequence 6: $1, \sqrt{2}, \sqrt{3}, 2, \sqrt{5}, ...$ (The pattern consists of square roots of consecutive integers: $\sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, ...$).

Key Terms and Definitions

• Sequence: A list of numbers arranged in a specific order, following a rule.
  • Example: $3, 6, 12, 24, ...$
• Term: Each individual number within a sequence.
  • Example: In the sequence $3, 6, 12, 24$, the numbers 3, 6, 12, and 24 are all terms.
• Position (or Index): The specific place of a term in the sequence, typically counted as the 1st term, 2nd term, and so forth.
• Notation:
  • $a_1$ represents the first term.
  • $a_2$ represents the second term.
  • $a_n$ denotes the $n$-th term (the general term at position $n$).
• Rule (or Pattern Rule): A description of how terms are formed or how they change from one to the next.
  • Example: "Add 2" in the sequence $2, 4, 6, 8, ...$
  • Example: "Multiply by 2" in the sequence $3, 6, 12, 24, ...$
  • Example: "Squares" in the sequence $1, 4, 9, 16, 25, ...$
• Fibonacci Sequence: A specific type of sequence where each term after the first two is the sum of the two preceding terms.
  • Example 1: $1, 1, 2, 3, 5, 8, 13, ...$
  • Example 2: $2, 2, 4, 6, 10, 16, ...$
  • Example 3: $5, 5, 10, 15, 25, 40, ...$

Classification of Sequences

• Finite Sequence: A sequence that has a specific, countable number of terms. It has a definite beginning and an end, stopping after a certain point.
  • Example: $2, 4, 6, 8, 10$
  • This specific sequence has 5 terms and ends exactly at 10.
• Infinite Sequence: A sequence that has no end and continues forever. It follows a rule endlessly.
  • Example: $1, 2, 4, 8, 16, 32, ...$
  • This sequence keeps doubling and never terminates.
  • Notation for Infinity: An ellipsis "…" is used to show that the sequence continues without end.

Advanced Sequence Examples and Strategies

• Constant Addition:
  • Sequence: $5, 9, 13, 17, ...$
  • Strategy: Identify differences. In this case, $+4$ each time.
  • Next Term Calculation: $17 + 4 = 21$
• Constant Multiplication:
  • Sequence: $3, 6, 12, 24, ...$
  • Strategy: Multiply by a constant (2).
  • Next Term Calculation: $24 \times 2 = 48$
• Prime Increments:
  • Sequence: $9, 12, 17, 24, 35, ...$
  • Strategy: Add consecutive prime numbers ($+3, +5, +7, +11, ...$).
  • Next Term Calculation: $35 + 13 = 48$
• Fibonacci-style Addition:
  • Sequence: $1, 1, 2, 3, 5, 8, ...$
  • Strategy: Add the previous two terms.
  • Next Term Calculation: $5 + 8 = 13$
• Interleaved Sequences:
  • Sequence: $2, 10, 4, 12, 6, 14, ...$
  • Strategy: Two patterns occurring in alternating positions. Odd positions ($2, 4, 6$) increase by 2. Even positions ($10, 12, 14$) increase by 2.
  • Next Term Calculation: Following the odd position pattern after 14, the next term is 8.
• Alternating Operations:
  • Sequence: $2, 5, 10, 13, 26, 29, ...$
  • Strategy: Alternate between adding 3 and multiplying by 2 (Rule: $+3, \times 2, +3, \times 2, +3$).
  • Next Term Calculation: $29 \times 2 = 58$
• Growing Differences (Odd Numbers):
  • Sequence: $2, 5, 10, 17, 26, ...$
  • Strategy: Differences are increasing odd numbers ($+3, +5, +7, +9$).
  • Next Term Calculation: $26 + 11 = 37$
• Fractional Patterns:
  • Sequence: $\frac{1}{2}, \frac{3}{4}, \frac{9}{8}, \frac{27}{16}, ...$
  • Strategy: Manage numerator and denominator separately. Numerator is multiplied by 3; denominator is multiplied by 2.
  • Next Term Calculation: $\frac{27 \times 3}{16 \times 2} = \frac{81}{32}$
• Negative Ratio (Sign Flip):
  • Sequence: $16, -8, 4, -2, 1, ...$
  • Strategy: Multiply by a negative ratio of $- \frac{1}{2}$ each step.
  • Next Term Calculation: $1 \times - \frac{1}{2} = - \frac{1}{2}$
• Triangular Numbers:
  • Sequence: $1, 3, 6, 10, 15, ...$
  • Strategy: Add increments of 1, 2, 3, 4, 5…
  • Next Term Calculation: $15 + 6 = 21$

Arithmetic Sequences

• Definition: A specific type of number pattern where any two consecutive terms have a constant difference, known as the common difference ($d$).
• Common Difference ($d$) Examples:
  1. $4, 8, 12, 16, ...$ where $d = 4$
  2. $-5, -2, 1, 4, ...$ where $d = 3$
  3. $-8.5, -5.5, -2.5, 0.5, ...$ where $d = 3$
• Arithmetic Sequence Formula:     The $n$-th term of an arithmetic sequence is found using the formula:     $a_n = a_1 + (n - 1)d$
  • $a_n$ = $n$-th term
  • $a_1$ = first term
  • $n$ = position of the term
  • $d$ = common difference

Arithmetic Sequence Practice Problems

• **Example 1: Find the 20th term in the sequence $6, 13, 20, 27, ...$

  • Identify $d$: $13 - 6 = 7$; $20 - 13 = 7$; $27 - 20 = 7$. Thus, $d = 7$.
  • Identify values: $a_1 = 6, d = 7, n = 20$
  • Apply formula: $a_{20} = 6 + (20 - 1)7$
  • Calculation: $a_{20} = 6 + (19)7 = 6 + 133 = 139$
  • Result: The 20th term is 139.

• **Example 2: Find the value of $n$ if $a_n = 75$ in the sequence $3, 9, 15, 21, ...$

  • Identify values: $a_1 = 3$, $d = 6$, $a_n = 75$
  • Apply formula: $75 = 3 + (n - 1)6$
  • Calculation:
    • $75 = 3 + 6n - 6$
    • $75 = 6n - 3$
    • $78 = 6n$
    • $n = 13$
  • Result: The value of $n$ is 13.

• Example 3: Solve for properties given two terms.

  • Given: $a_5 = 20$ and $a_{12} = 48$.
  • (a) Find Common Difference ($d$):$d = \frac{a_{12} - a_5}{12 - 5} = \frac{48 - 20}{7} = \frac{28}{7} = 4$
  • (b) Find First Term ($a_1$):         Use $a_5 = 20, n = 5, d = 4$$20 = a_1 + (5 - 1)4$$20 = a_1 + 16$$a_1 = 4$
  • (c) Find 30th Term ($a_{30}$):         Use $a_1 = 4, n = 30, d = 4$$a_{30} = 4 + (30 - 1)4 = 4 + (29)4 = 4 + 116 = 120$

Geometric Sequences

• Definition: Also known as a geometric progression, it is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).
• Common Ratio ($r$):
  • Can be determined by dividing any term by the term that precedes it.
  • If $r > 1$, the sequence grows.
  • If $0 < r < 1$, the sequence shrinks.
  • If $r < 0$, the sequence alternates between positive and negative values.
• Geometric Sequence Formula:     The $n$-th term is given by:     $a_n = a_1 r^{n-1}$     where $r \neq 0$.

Geometric Sequence Practice Problems

• Example 1: Identify ratio and next term

  • $1, 2, 4, 8, ...$ (Ratio $r = 2$). Next term: $8(2) = 16$.
  • $80, 20, 5, ...$ (Ratio $r = \frac{20}{80} = \frac{1}{4}$). Next term: $5(\frac{1}{4}) = \frac{5}{4}$.
  • $2, -8, 32, -128, ...$ (Ratio $r = \frac{-8}{2} = -4$). Next term: $-128(-4) = 512$.

• **Example 2: Find the 8th term of $3, 6, 12, 24, ...$

  • Identify values: $a_1 = 3, r = 2, n = 8$
  • Apply formula: $a_8 = 3 \times 2^{8-1} = 3 \times 2^7$
  • Calculation: $a_8 = 3 \times 128 = 384$

• **Example 3: Find the 6th term of $100, 50, 25, ...$

  • Identify values: $a_1 = 100, r = 0.5, n = 6$
  • Apply formula: $a_6 = 100 \times (0.5)^{6-1} = 100 \times (0.5)^5$
  • Calculation: $a_6 = 100 \times 0.03125 = 3.125$

• **Example 4: Find the 7th term of $2, -10, 50, -250, ...$

  • Identify values: $a_1 = 2, r = -5, n = 7$
  • Apply formula: $a_7 = 2 \times (-5)^{7-1} = 2 \times (-5)^6$
  • Calculation: $a_7 = 2 \times 15625 = 31250$

Real-World Application: Savings Challenge

• Scenario: A student joins a savings challenge for a Christmas gift. He saves $P5$ on the first day and doubles the amount every day after that.
• Objective: Find the amount saved on the 10th day alone.
• Identification: This is a geometric sequence: $5, 10, 20, ...$
  • $a_1 = 5$
  • $r = 2$
  • $n = 10$
• Solution:
  • Formula: $a_{10} = 5 \times 2^{10-1}$
  • Calculation: $a_{10} = 5 \times 2^9 = 5 \times 512 = 2560$
• Conclusion: He will need to save $P2,560$ on the 10th day.

Quiz and Exercise Data

• Arithmetic Sequence Tasks: Find common difference ($d$), explicit formula, and the specific term.

  • $12, 16, 20, 24, ...$ (Find $a_{31}$)
  • $0, -4, -8, -12, ...$ (Find $a_{30}$)
  • $1, 5, 9, 13, ...$ (Find $a_{10}$)
  • $13, 9, 5, 1, ...$ (Find $a_{10}$)
  • $-8, -5, -2, 1, 4, ...$ (Find $a_{12}$)
  • $2, 11, 20, ...$ (Find $a_7$)
  • $9, 6, 3, ...$ (Find $a_8$)
  • $5, 9, 13, 17, ...$ (Find $a_{15}$)
  • $2, 6, 10, ...$ (Find $a_6$)

• Geometric Sequence Tasks: Find common ratio ($r$), explicit formula, and the specific term.

  • $4, 8, 16, 32, ...$ (Find $a_{10}$)
  • $4, 16, 64, 256, ...$ (Find $a_9$)
  • $-1, -2, -4, -8, ...$ (Find $a_8$)
  • $-2, 6, -18, 54, ...$ (Find $a_{12}$)
  • $-2, 10, -50, 250, ...$ (Find $a_7$)