Sequences & Series

Sequences

• A sequences is a series of numbers written following a simple rule.• 2, 5, 8, 11, 14,…

• 3, 6, 12, 24, 48….

• 1, 1, 2, 3, 5, 8, 13…

• 1, - ½ , ¼ , - 1/8 …

• 1, 2, 3, 1, 2, 3, 1, 2, 3…

• 1, -1, 1, -1, 1….

• If the n th term, u of a sequence is known, then all the terms of the sequences can be found.

• A finite sequence has a fixed number of terms.

• A infinite sequence has an infinite number of terms.

Example 3.1

• Find the first four terms of the sequence whose nth term is given by u = 32 – 6n

Example 3.2

• Find the first four terms of the sequence where

un = (-3)n

Example 3.3

• Find the first four terms of the sequence which is defined by u n + 1 = 2un + 1, n > 1 where u1 = 1

Exercise

• Find the first five terms of the sequence given in the following.• un = 3un-1 and u1 = 2

• u n + 1 = 5un – 6un-1 and u1 = 5, u2 = 13

Arithmetic Sequence - AS (Progression)

• There is a common difference between successive terms.

• Example:• 2, 6, 10, 14….

• 21, 18, 15, 12…

• a , a + d , a + 2d ,…..

General terms of A.S

• AS has first term, a and common difference d

• The sequence isa , a + d , a + 2d , a + 3d …

First term , T1 = a

Second term, T2 = a + d = a + (2-1)d

Third term, T3 = a + 2d = a + (3-1)d

n th term, Tn = a + d = a + (n-1)d

Example 3.4

• Find,• The 20th term, and

• The nth term of the following A.S

a. 3, 7, 11, 15, …..

b. 13, 10, 7, 4, …..

Example 3.5

• Find the number of terms in the A.S

7, 13, 19, ………, 307

Example 3.6

• How many multiples of 7 are there between 100 and 3000?

Example 3.7

• The 6th term of an A.S is 22 and the 10th term is 38

Find,

a. The 1st term and the common difference

b. The 100th term

Example 3.8

• The nth term of an A.S is 40 + 7n. Find the common difference.

• Which term of the sequence is 215?

Exercise

Find,

i. the 1st term

ii. the common differencea. The 2nd term is 3 and the 8th term is -21

b. The 8th term is 11 and 15th term is 32

Exercise

• How many multiples of 7 between 200 and 5000?

Exercise

• If 9, x, y, z, 21 form of A.S, find the values of x, y, z.

Exercise

• The sum of the 3rd and 4th of an A.S is twice the 2nd

term and the 5th term exceeds the 1st term by 10. find the 1st term and common difference.

Arithmetic Series (Sn)

• Arithmetic Series is the sum of the terms of an A.S

• Sum of n th term of an Sn

Sn =n

2( a + l )

Sn =n

2[ 2a + (n-1)d ]

Tn = Sn – Sn-1

Example 3.9

• Find the sum of following series.

a. An arithmetic series of 25th terms where the 1st

term is 5 and the last term is 41

b. 3 + 5 + 7 + …… to 60 terms

Example 3.10

• Find an expression for the sum of the first n th term of the series 5 + 8 + 11 + …

• Find the value of n if the sum is equal to 440.

Example 3.11

• Find at least number of term required for the sum of the Sn = 7 + 13 + 19 + ….. to exceed 1000.

Example 3.13

• The sum of the first term 15 terms of Sn is 255 and the sum of the next 15 terms is 705. Find the 1st

term and the common series.

Example 3.14

• Find the sum of all the positive integers less than 200 which are:

a. multiples of 3 or 7 or both,

b. not multiples of 3 or 7

Exercise

• The 1st term of Sn is 16 and the 6th term is 83. Find the 3rd term and the sum of the 1st 40 terms of the sequence.

Exercise

• Find the sum of all the integers between 200 and 400 that are divisible by 7.

Exercise

• Find the sum of all the positive integers less than 200 which are:

a. multiples of 5 or 7 or both,

b. not multiples of 5 or 7