Arithmetic and Geometric Sequences and Series

• A sequence is an ordered list of numbers.
• A series is the sum of the terms in a sequence.

Arithmetic Sequences and Series

• An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
• This constant difference is called the common difference, denoted by 'd'.
• The general form of an arithmetic sequence is: a, a + d, a + 2d, a + 3d,... , where 'a' is the first term and 'd' is the common difference.
• The n-th term (a_n) of an arithmetic sequence can be expressed as a_n = a_1 + (n-1)d, where a_1 is the first term and n is the term number. Alternatively, a_n = a + (n - 1)d where a is the first term.
• The sum of the first n terms (S_n) of an arithmetic series is given by S_n = n/2 * (a_1 + a_n) or S_n = n/2 * [2a_1 + (n-1)d]. Alternatively, S_n = n/2 * [2a + (n - 1)d] where a is the first term.
• Given any two terms in an arithmetic sequence, a_1 and d can be determined.
• An arithmetic series is the sum of the terms in an arithmetic sequence.
• Example: In the arithmetic sequence 2, 5, 8, 11,..., the first term (a) is 2 and the common difference (d) is 3.
• Example: To find the sum of the first 10 terms of the arithmetic sequence 2, 5, 8, 11,..., where a = 2, d = 3, and n = 10: S10 = 10/2 * [2(2) + (10 - 1)(3)] = 5 * [4 + 27] = 5 * 31 = 155.

Geometric Sequences and Series

• A geometric sequence is a sequence where the ratio between consecutive terms is constant.
• This constant ratio is called the common ratio, denoted by 'r'.
• The general form of a geometric sequence is: a, ar, ar^2, ar^3,..., where 'a' is the first term and 'r' is the common ratio.
• The n-th term (a_n) of a geometric sequence can be expressed as a_n = a_1 * r^(n-1), where a_1 is the first term and n is the term number. Alternatively, a_n = a * r^(n-1) where a is the first term.
• The sum of the first n terms (S_n) of a geometric series is given by S_n = a_1 * (1 - r^n) / (1 - r), where r ≠ 1. Alternatively, S_n = a * (1 - r^n) / (1 - r), where r ≠ 1 and a is the first term.
• If |r| < 1, the sum of an infinite geometric series converges to S = a_1 / (1 - r). Alternatively, S = a / (1 - r), where |r| < 1 and a is the first term.
• If |r| ≥ 1, the sum of an infinite geometric series does not converge (it diverges).
• Given any two terms in a geometric sequence, a_1 and r can be determined.
• A geometric series is the sum of the terms in a geometric sequence.
• Example: In the geometric sequence 3, 6, 12, 24,..., the first term (a) is 3 and the common ratio (r) is 2.
• Example: To find the sum of the first 5 terms of the geometric sequence 3, 6, 12, 24,..., where a = 3, r = 2, and n = 5: S5 = 3 * (1 - 2^5) / (1 - 2) = 3 * (1 - 32) / (-1) = 3 * (-31) / (-1) = 93.
• An infinite geometric series is a geometric series with an infinite number of terms.
• The sum of an infinite geometric series (S∞) exists only if the absolute value of the common ratio (|r|) is less than 1 (|r| < 1).
• Example: Consider the infinite geometric series 1 + 1/2 + 1/4 + 1/8 +... Here, a = 1 and r = 1/2. Since |1/2| < 1, the sum exists: S∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2.

Applications of Arithmetic and Geometric Sequences

• Arithmetic sequences can model situations with constant increases or decreases.
• Geometric sequences can model situations with exponential growth or decay.
• Compound interest calculations are an example of a geometric sequence.
• Arithmetic sequences and series can be used to model situations with a constant rate of change, such as simple interest, linear depreciation, and evenly spaced quantities.
• Geometric sequences and series can be used to model situations with exponential growth or decay, such as compound interest, population growth, and radioactive decay.

Key differences between arithmetic and geometric sequences

• Arithmetic sequences have a common difference between terms using addition/subtraction.
• Geometric sequences have a common ratio between terms using multiplication/division.
• Arithmetic sequences have a constant difference between terms; geometric sequences have a constant ratio.
• Arithmetic series involve addition of terms with a constant difference; geometric series involve addition of terms with a constant ratio.
• Infinite geometric series have a finite sum only if the absolute value of the common ratio is less than 1; arithmetic series do not have a finite sum if the common difference is not zero.

Sigma Notation

• Sigma notation (∑) is a concise way to represent the sum of a series.
• The general form is ∑_(i=m)^n f(i), where i is the index of summation, m is the lower limit, n is the upper limit, and f(i) is the expression to be summed.
• For example, ∑_(i=1)^5 i represents the sum 1 + 2 + 3 + 4 + 5.
• Properties of sigma notation:
  • ∑(i=1)^n (c * a_i) = c * ∑(i=1)^n a_i (Constant multiple)
  • ∑(i=1)^n (a_i + b_i) = ∑(i=1)^n a_i + ∑_(i=1)^n b_i (Sum of terms)
• Sigma notation (Σ) is a concise way to represent the sum of a sequence or series.
• The general form of sigma notation is: Σ (expression) from i = start to end, where 'i' is the index of summation, 'start' is the starting value of the index, and 'end' is the ending value of the index. The 'expression' defines the terms to be summed, usually involving 'i'.
• Example: The sum of the first 5 natural numbers (1 + 2 + 3 + 4 + 5) can be written in sigma notation as: Σ i from i = 1 to 5.
• Example: The series 2 + 4 + 6 + 8 can be written as Σ (2i) from i=1 to 4

Special Series

• Sum of the first n natural numbers: ∑_(i=1)^n i = n(n+1)/2.
• Sum of the squares of the first n natural numbers: ∑_(i=1)^n i^2 = n(n+1)(2n+1)/6.
• Sum of the cubes of the first n natural numbers: ∑_(i=1)^n i^3 = [n(n+1)/2]^2.