Sequences: Algebra lesson

Introduction to Sequences

• The first lesson for the second year of secondary school is introduced.
• The complete curriculum will be uploaded to YouTube on a fixed schedule throughout the semester.
• Viewers are encouraged to subscribe and like the video.
• The curriculum includes Algebra and Trigonometry as in the previous semester.
• Calculus will cover limits and differential calculus.
• Algebra will cover sequences and permutations.
• Students entering mathematical science are strongly advised to focus on mathematics as preparation for the third year, as it builds upon previous knowledge.

Understanding Sequences

• Sequences are introduced through examples.
• A simple arithmetic sequence is: 2, 4, 6, 8, which can be continued by adding 2 to each term.
• Another example includes 1, 3, 5, 7 where each number increases by 2.
• A more complex example involves alternating signs and fractions: 1, -1/2, 1/4, -1/8, 1/16.
• Another example: 1/1, 2/4, 3/9, 4/16 can be continued by incrementing the numerator and squaring it for the denominator, next term 5/25.

Terminology and General Terms

• Sequences will be discussed more formally in this term.
• A sequence like 2, 4, 6, 8, 10 is used to explain the concepts.
• "Tₙ" represents the entire sequence.
• T₁ refers to the first term, T₂ to the second, and so on.
• "Tₙ" specifically refers to the general term.
• The general term is the formula that can produce any term in the sequence.
• Questions will either ask to find terms from a general term or vice versa.

Types of Sequences

• Sequences can be increasing or decreasing.
• An increasing sequence gets larger, e.g., each term is greater than the previous one.
• A decreasing sequence gets smaller, such as: 30, 25, 20, 15.

Finding Terms from a Given General Term

• Given a general term, specific terms of the sequence can be found.
• To find the first five terms, substitute n = 1, 2, 3, 4, 5 into the general term formula.
• Example is provided where Tₙ = 2n - 3 to find first five terms.
• The process involves substituting values 1 through 5 into the general term formula.
• The resulting sequence is then written out explicitly.
• Another example with Tₙ = (-1)ⁿ / (n + 3).
• Calculate terms by substituting n = 1, 2, 3, 4, 5 into the general term.
• The resulting sequence is written out.
• Another example: Tₙ = 1 + 2/√n. Terms calculated: 2, 1 + 2/√2, 1 + 2/√3, 2, 1 + 2/√5.

Recursive Sequences

• A different type of sequence is introduced where each term depends on the previous one.
• Example: given T₁ = 2, and Tₙ₊₁ = 2 * Tₙ.
• This means each term is twice the previous term.
• Starting with T₁ = 2, the subsequent terms are calculated as T₂ = 4, T₃ = 8, T₄ = 16, T₅ = 32.
• Each term is found by multiplying the previous term by 2.

Finding the General Term from a Given Sequence

• How to find the general term of a sequence if it alternates signs is explained.
• If a sequence alternates signs, include (-1)ⁿ in the general term.
• If the sequence starts with a negative term, use (-1)ⁿ.
• If the sequence starts with a positive term, use (-1)ⁿ⁺¹.
• If terms increase by a constant value, the general term may involve multiplication by n.
• The general term for the sequence 2, 4, 6, 8 is Tₙ = 2n.
• Some sequences are transformations of known sequences.
• Recognising standard sequences helps determine the general term.

Examples of Finding General Terms

• Example: 1, 4, 9, 16.
• These are recognized as squares.
• The general term is Tₙ = n².
• Another example: 3, 9, 27, 81.
• These are recognized as powers of 3 and exponents start from 1 hence Tₙ = 3ⁿ.
• Another example involves powers of 3 but exponents start from zero hence Tₙ = 3ⁿ⁻¹.

Using Multiple Choice to Find the General Term

• If unsure, substitute the values into multiple-choice options.
• Substitute n = 1, 2, 3, and so on into each general term to see if it matches the sequence.

Examples of Finding the General Term Using YouTube Style

• A YouTube-like method for Tₙ = n / (n + 1) is shown,.
• Substitute n = 1, 2, 3, 4 to see if the general term matches the sequence.
• The first term would be 1 / (1 + 1) = 1/2.
• The second term would be 2 / (2 + 1) = 2/3.
• Compare the calculated terms with the given sequence to select the correct general term.
• Another example: 1/2, 1/6, 1/12.
• Test options, such as Tₙ = 1 / (n² + n).
• This concludes the first lesson.
• Viewers are encouraged to like and subscribe.

Introduction to sequences with examples such as arithmetic sequences and alternating sign sequences. Includes examples like 2, 4, 6, 8 and 1, -1/2, 1/4, -1/8, 1/16. Important concepts for students in the second year of secondary school.