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 number 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 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.

Arithmetic Mean

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

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.

Solution

(i)         Let the sequence be -5, a, b, c, d, 10.

                        a = -5, T6 = 10, n =6.

                        T_n = a + (n-1) d

                        10 = -5 + (6 – 1) d

                        15 = 5d

                        d = ¹⁵/₅ = 3

            a = -5 + 3 = -2

            b = -2 + 3 = 1

            c = 1 + 3 = 4

            d = 4 + 3 = 7

The numbers will be -5, -2, 1, 4, 7, 10.

(ii)         T₈ = a + 7d = 18, T₁₂ = a +11d = 26

       a + 7d = 18    ……………….. (i)       

            a + 11d =26   ……………….. (ii)

            Subtract (i) from (ii)

            4d = 8

            d = 2

            Substitute for d = 2 in (i)

            a + 7 (2) = 18

            a = 18 – 14

            a = 4

            T₂₀ = a + (n – 1) d   = a + 19d

            T₂₀ = 4 + (20 – 1) 2

                    = 4 + 19 x 2             T₂₀   = 42

Evaluation

(1)        Given that 4, p, q, 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)

Evaluation:

The sum of the first ten term of a linear sequence is -60 and the sum of the first fifteen term of the sequence is -165. Find the 18^th term of the sequence.

General Evaluation

1. The sum of the first four terms of a linear sequence (A.P) is 26 and that of the next four terms is 74.

     Find the values of (i) the first term (ii) the common difference.

2. Calculate the (i) common difference (ii) the 20^th term of the arithmetic progression;

     100, 96, 92, 88, 86…

3.  Solve the equation:  log₄(x² + 6x + 11) = ½

Reading Assignment: Further Mathematics Project Book 1(New third edition).Chapter 28 -33 & 36 – 37

Weekend Assignment

1.          Find T₉ of the sequence -1, 2, 5, 8 …………….    A.   21   B.  22        C.  23       D.  24

2.          The 10^th term of an A.P is 68 and the common difference is 7, find the first term of the sequence.  

A.   3     B.   5         C.   7 D. 9

3.          Find the sum of the first twelve term of the sequence 2, 5, 8, 11…   A. 202   B. 212  C.222 D. 232

4.          What is the general term of the sequence 31, 26, 21, 16, 11…………

            A.   1 + 4n       B. 3 x 2^n-1          C. 36 – 5n     D. 5(½)^n-2

S_n = ⁿ/₂ (a + T_n)

But T_n = a + (n-1) d

S_n = ⁿ/₂ (2a + (n-1) d)

Theory

1.          The first three terms of an A.P are x, 2x+1, 4x+1, find x and the sum of the first 18 terms.

2.          The sum of the first twenty –one terms of an A.P is 28, and the sum of the first twenty-eight terms is 21. Find which terms of the sequence is o and also the sum of the term proceeding it.