Sequences and Series Study Notes

Grade 12 Sequences and Series Revision Notes

Overview

These notes cover sequences and series as revised from Grade 10 and 11, focusing on arithmetic sequences, geometric sequences, and their formulas, examples, and applications in real life, particularly illustrating the concepts through the growth of Californian Redwood trees.

Arithmetic Sequences

  • Definition: An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant.
  • General term formula: T_n = a + (n - 1)d Where:
    • $T_n$: n-th term of the sequence
    • $a$: first term
    • $d$: common difference
    • $n$: position of the term

Example

  • Given a sequence: 40, 37, 34, …, find the common difference:
    • $d = T2 - T1 = 37 - 40 = -3$
    • First term $a = 40$
    • General term then is:
      T_n = 40 + (n - 1)(-3)

Summing Terms of an Arithmetic Series

  • Formula for the sum of the first n terms:
    Sn = rac{n}{2} (2a + (n - 1)d) or Sn = rac{n}{2} (a + b)
    Where $b$ is the last term - the n-th term.

Example

  • Calculate the sum of 4 + 7 + 10:
    • First term $a = 4$, second term $T_2 = 7$ gives $d = 7 - 4 = 3$.
    • The fourth term $T_4 = 4 + (4 - 1)(3) = 13$.
    • Summing gives:
      S_n = rac{4}{2} (4 + 13) = 2 imes 17 = 34

Geometric Sequences

  • Definition: A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • General term formula: T_n = ar^{n-1} Where:
    • $T_n$: n-th term
    • $a$: first term
    • $r$: common ratio
    • $n$: position of the term

Example

  • Given a sequence: 2, 4, 8, 16, …, find the common ratio:
    • $r = rac{4}{2} = 2$. The formula becomes:
      T_n = 2(2^{n-1})

Summing Terms of a Geometric Series

  • Formula for the sum of the first n terms:
    S_n = a rac{1 - r^n}{1 - r} ext{ (if } r
    eq 1 ext{)}

Example

  • Calculate the sum of the geometric series 1 + 2 + 4 + … + 16:
    • Here, $a = 1$, $r = 2$, and $n = 5$ (as there are 5 terms).
      S_n = 1 rac{1 - 2^5}{1 - 2} = 1 rac{1 - 32}{-1} = 31

Sigma Notation

  • Definition: A compact way to represent the sum of a sequence:
    Sn = ext{ } ext{Σ}{k=m}^{n} Tk Indicates summing function $Tk$, from $k = m$ to $n$.

Example

  • For the terms $T_k = 5k - 3$ for $k = 1$ to $k = 7$:
    S = (5(1) - 3) + (5(2) - 3) + (5(3) - 3) + (5(4) - 3) + (5(5) - 3) + (5(6) - 3) + (5(7) - 3)

Applications in Real Life

Growth of Californian Redwood Trees

  • Californian Redwood trees can live longer than 2000 years and grow on average by 2.4 meters in their first 50 years and then grow in height by 30% of the previous year's growth thereafter. They theoretically never exceed a height of 122 meters.

Calculations

(a) Average height after 50 years:

  • $50 ext{ (years)} imes 2.4 ext{ (meters per year)} = 120 ext{ meters}$

(b) Growth calculation for year 51 onwards:

  1. Growth in year 50 = 2.4 meters
  2. Growth in year 51 = $0.3 imes 2.4 = 0.72 ext{ meters}$
  3. Growth for ensuing years continues multiplying by 0.3.

Proving Height Limit

  • Total growth contribution from year 1 to 50 + growth from year 51 to 70:
    = 120 + (0.72 + 0.216 + 0.0648 + … ext{ for 20 terms})
  • This is a geometric series with first term $a = 0.72$ and ratio $r = 0.3$. Using
    S = rac{a(1 - r^{20})}{1-r} we calculate the limit, eventually showing the trees will never exceed 122 meters.

Simultaneous Equations in Sequences

  • A method to solve problems in arithmetic sequences where:
  1. A term position and its value is given (e.g., 5th term = 12).
  2. Use simultaneous equations to find values of a and d:
    T_n = a + (n - 1)d