SEQUENCES AND SERIES

Introduction - Sequence A sequence or progression is an ordered set of numbers which can be generated from a rule. General sequence terms as denoted as follows a1 – first term , a2 – second term, …, an – nth term etc The rule may give nth term, an, as a function of n

Introduction – Sequence (cont) Example: Given a infinite sequence The rule or general term  the general term is

Exercises: Write down the first four term of the sequence with general term What is the twenty-first term of the sequence? What is the 100th term of the sequence of

Arithmetic Sequence (Progression) -AP An arithmetic sequence or arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a common constant. In general, an AP is written in the form of {a, a + d, a + 2d, a + 3d,…a + (n – 1)d,…} where a is the first term and d is the common difference

Arithmetic Sequence (Progression) –AP (cont) If the initial term of an AP is a1 and the common difference of successive numbers is d, then the nth term of the sequence is given by: Example 1: Write down the nth term of the arithmetic sequences -10, -5, 0, 5,…

Arithmetic Sequence (Progression) –AP (cont) Example 2: The 4th term of an arithmetic sequence is 12 and the tenth term is 42. Given that the first term is u1 and the common difference is d, write down two equations in u1 and d that satisfy this information. Solve the equations to find the values of u1 and d.

Geometric Sequences (Progression) – GP A geometric progression (GP) is a sequence of numbers in which each number is multiplied by the same factor to obtain the next number in the sequence. For example, the sequence 2, 6, 18, 54,… is a GP with common ratio 3. Thus, the general form of a GP is where a is the first term and r is the common ratio

Geometric Sequences (Progression) – GP (cont) The nth term of a GP with initial value a and common ration r is given by Note: the common ratio r should not be 0 or 1 If r is 0, then we have : a, 0, 0, 0, … (not GP) If r is 1, then we have : a, a, a, a, … (not GP) Example 3: Write down the first five terms of the GP which has first term 1 and common ration ½ . Find the 10th and 20th term of the sequence.

Geometric Sequences (Progression) – GP (cont) Example 4: The 3rd term of a GP is 3 and the 6th term is 3/8. Find the common ratio r and the first term a1.

Limit of infinite sequence The limit of a sequence is the value to which its term approach indefinitely as n becomes large. If the limit of a sequence an is L, we can write as If a sequence has a (finite) limit, then it is said to be convergent. If a sequence becomes arbitrarily large (approaches ), then it is said to be divergent

Limit of infinite sequence (cont) Example 5:

Series and Partial Sums When we sum up just n terms of a sequence, it is called the partial sums Sn. When we sum up an infinite sequence, it is called a series .

Arithmetic Series Given an arithmetic sequence Then, the arithmetic series can be written as The sum of the first n terms of an arithmetic sequence is where a1 is the first term and l is an , the last term of a finite sequence. Or

Arithmetic Series (cont) Example 6: Find the sum of the first 50 terms of the sequence {1, 3, 5, 7, 9, …} Find the sum of the series 1+ 3.5 + 6 + 8.5 + …+ 101

Geometric Series Given a geometric sequence Then, the geometric series can be written as The sum of the first n terms of an geometric sequence is provided that r  1.

Geometric Series (cont) Example 7: Given the first two terms of a geometric progression as 2 and 4, what is the sum of the first 10 terms? Find the sum of the first 20 terms of the geometric series 2 + 6 + 18 + 54 + …

Convergence /divergence series The nth partial sum of an arithmetic series is What will happen when n becomes very large (approaches )? That is The series approaches infinity. Thus, an arithmetic series is a divergence series for any a and d.

Convergence /divergence series (cont) The nth partial sum of an geometric series is What will happen when n becomes very large (approaches ) ? That is If|r|<1, then rn 0 as n , thus the series is converged i.e. If|r| > 1, then rn as n , thus the series is diverged

Convergence /divergence series (cont) Example 8: For the geometric progression whose first two terms are 5 and ½ , find S. Consider a geometric progression whose first three terms are 12, -6, 3. Find both S8 and S. For the geometric progression whose first two terms are 2 and 4, find S7, S20 and S.