Sequences

Understanding Sequences

  • A sequence is an ordered list of numbers. Each number in the sequence is called a ‘term’.

  • A sequence can be either finite, with a specific number of terms, or infinite, continuing indefinitely.

  • Sequences can be defined by a rule, which describes the pattern of the terms.

Arithmetic Sequences

  • An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant is known as the common difference.

  • An arithmetic sequence can be expressed using the formula: a + (n-1)d, where ‘a’ is the first term, ‘n’ is the term number, and ‘d’ is the common difference.

Geometric Sequences

  • A geometric sequence is a sequence of numbers where the ratio between any two consecutive terms is constant. This constant is referred to as the common ratio.

  • A geometric sequence can be expressed using the formula: ar^(n-1), where ‘a’ is the first term, ‘r’ is the common ratio, and ‘n’ is the term number.

Finding the nth Term

  • To find the nth term of an arithmetic sequence, use the formula: nth term = a + (n-1)d.

  • To find the nth term of a geometric sequence, use the formula: nth term = ar^(n-1).

Sum of Sequence Terms

  • The sum of the first ‘n’ terms of an arithmetic sequence can be found with the formula: S_n = n/2(2a + (n - 1)d) where S_n is the sum of the first n terms, ‘a’ is the first term, ‘n’ is the number of terms, and ‘d’ is the common difference.

  • The sum of the first ‘n’ terms of a geometric sequence can be calculated using the formula: S_n = a(1 - r^n) / (1 - r) for r ≠ 1, where S_n is the sum of the first ‘n’ terms, ‘a’ is the first term, ‘r’ is the common ratio and ‘n’ is the number of terms.

Real-World Applications of Sequences

  • Arithmetic and geometric sequences often show up in real-world scenarios. For example, arithmetic sequences can represent savings plans where a fixed amount is deposited every month.

  • Geometric sequences, on the other hand, can represent growth patterns or compound interest, where the growth or interest is a fixed percentage of the current amount.