SEQUENCE & SERIES

FURTHER MATHEMATICS SSS1 SECOND TERM

WEEK 1

SEQUENCE & SERIES

Content

• Sequence and series
• Arithmetic Progression (AP)
• Arithmetic Mean
• Sum of terms in an AP

Sequence & Series

A sequence is a pattern of numbers arranged in a particular order. Each of the numbers in the sequence is called a term. The terms are related to one another according to a well-defined rule.

Consider the sequence 1, 4, 7, 10, 13 …., 1 is the first term,(T₁) 4 is the second term(T₂), 7 is the third term (T₃).

The sum of the terms in a sequence is regarded as a series. The series of the above sequence is

1 + 4 + 7 + 10 + 13 = 35

The nth term of a Sequence

The nth term of a sequence whose rule is stated may be represented by T_nso that T₁, T₂, T₃etc represent the first term, second term, third term … etc respectively.

Consider the sequence 5, 9, 13, 17, 21 ……..

T₁ = 5 + 4(0)

T₂ = 5 + 4(1)

T₃ = 5 + 4 (2)

T₄ = 5 + 4 (3)

T_n = 5 + 4 (n – 1)

T_n = 5 + 4n – 4       = 4n +1

when n = 30

T₃₀ = 4(30) + 1

T₃₀ = 121

Find the nth term of these sequences:

1. 3, 5, 7, 9 …… 2n + 1       

(ii) 0, 1, 4, 9 ……… (n -1)²    

(iii) 1/3, 3/4, 1, 7/6 ………………2n - 1

                                                                n + 2

Examples

Write down the first four terms of the sequence whose general term is given by:

(i) T_n = n+1                (ii) T_n = 5 x (¹/₂)^n-2

3n +2

Solution

i.        T_n     =   n+1

                      3n + 2

          T₁     =  1 + 1       = ²/₅

                      3(1) + 2    

                T₂     =    2 + 1        = ³/₈

                      3(2) +2

          T₃    =   3 + 1        = ⁴/₁₁

                      3(3) + 2 

          T₄    =   4 + 1        = ⁵/₁₄     

(ii)  T_n = 5 x (1/2)^n-2

T₁   = 5 x (1/2)^1-2   = 5(¹/₂)^-1  = 5(2^-1)^-1 = 5 x 2 = 10

T₂   = 5 x (1/2)^2-2    = 5(¹/₂)⁰  = 5 x 1 = 5

T₃   = 5 x (1/2)^3-2   = 5 x (¹/₂) = ⁵/₂

T₄   = 5 x (1/2)^4-2   = 5(¹/₂)²   = ⁵/₄

The sequence is 10, 5, ⁵/₂, ⁵/₄ ………

                 3(4) + 2

The sequence is ²/₅, ³/₈, ⁴/₁₁, ⁵/₁₄ ……..

Evaluation

Find the first term of the sequence whose general term is given by

(i) 50 – (½)ⁿ        (ii) 2 + ³/₂⁽ⁿ⁺¹⁾

Arithmetic Progression (A.P) or  Linear Sequence

An arithmetic progression (A.P) is generated by adding or subtracting a constant number to a preceding term to get a term. This constant number is called the common difference designated by the letter d. The first term is designated by a.

Examples    

1. What is the 10^th term of the sequence 10, 6, 2, -4 …..

2. Find the term of the A.P 3½, 7, 10½ ….. Which is 77.

3. The first term of an A.P is 3 and the 8^th term is 31. Find the common difference.

Solution

(1.)     The A.P   = 10, 6, 2, -4

          a = 10, d = 6 – 10 = - 4, n = 10

          T_n  = a + (n – 1) d

          T₁₀ = 10 + (10 – 1) (-4)

          T₁₀ = 10 +9(-4)   = 10 – 36

          T₁₀ = -26.

(2.)     A.P   = 3 ½, 7, 10½ ……………… 77

          a = 3½, d = 7 – 3½ = 3½, n =?  T_n = 77

          T_n = a + (n-1)d

          77 = 3½ + (n-1)3½

      77 = 3½ + 3½n - 3½

          77 = 3½ n

          n = ⁷⁷/_3½ = ⁷⁷/⁷/₂

          n = ⁷⁷ x ²/₇ = 22

(3)      a = 3, T₈ = 31, d = ? n = 8

T_n = a + (n-1) d

     31 = 3 + (8-1) d

          31 – 3 = 7d

          d = ²⁸/₇ = 4

Evaluation

(i)       Find the 15^th term of the A.P 5, 2, -1, -4 …………

(ii)      Find the term of the A.P 1, 6, 11, 16…. which is 66.

Arithmetic Mean

If a, b, c are three consecutive terms of an A.P, then the common difference, d, equal

b – a = c – b    = common difference.

b + b = a +c

2b = a + c

b = ½(a +c)

Examples

(i)       Insert four arithmetic means between -5 and 10.

(ii)      The 8^th term of a linear sequence is 18 and the 12^th term is 26. Find the first term, the common difference and the 20^th term.

(1)      Given that 4, p, q, and 13 are consecutive terms of an A.P, find the values of p and q.

(2)      The sum of the 4^th and 6^th terms of an A.P is 42. The sum of the 3^rd and 9^th terms of the progression is 52. Find the first term, the common difference and the twentieth term of the progression.

Sum of terms in an A.P

To find an expression for the sum of n terms of a linear sequence, Let S_n be the sum, then

S_n = a + (a + d) + (a + 2d) + ……. + T_n ………….. (i)

Also

S_n = T_n+ (T_n- d) + (T_n- 2d) + ……… a ………. (ii)

Adding (1) and (2)

2S_n = (a + T_n) + (a + T_n) + (a + T_n) + ………… + (a + T_n)

\2S_n = n (a + T_n)

\S_n = ⁿ/₂ (a + T_n)

But T_n = a + (n-1) d

S_n = ⁿ/₂ (2a + (n-1) d)

Examples

 (1) Find the sum of the first 25 terms of the A.P 3, 10, 17 ……….

 (2) Find the sum of the first eight terms of a linear sequence whose first term is 6 and the last term is 46.

 (3) The sum of the first ten terms of an arithmetic progression is 255. Find the sum of the next twenty

        term of the A.P if the sum of the first twenty terms is 1010.   

Solution