**The**

*n*th Term of Arithmetic Progression (Examples)

**Example 1:**

If the 20th term of an arithmetic progression is 14 and the 40th term is –6,

Find

Find

**(a)**the first term and the common difference,

**(b)**the 10th term.

*Solution:***(a)**

*T*

_{20}= 14

*a*+ 19

*d*= 14 —– (1) ←

**(**

*T*_{n}_{ }= a + (*n*– 1)*d**T*

_{40}= – 6

a + 39

*d*= – 6 —– (2)(2) – (1),

20

*d =*– 20

*d =***– 1**

Substitute

*d =*– 1 into (1),*a*+ 19 (– 1) = 14

*a***= 33**

(b)

(b)

*T*

_{10}=

*a*+ 9

*d*

*T*

_{10}= 33 + 9 (– 1)

*T*

_{10}**= 24**

Example 2:

Example 2:

The 3rd term and the 7th term of an arithmetic progression are 20 and 12 respectively.

**(a)**Calculate the 20th term.

**(b)**Find the term whose value is – 34.

*Solution:***(a)**

*T*

_{3}= 20

*a*+ 2

*d*= 20 —– (1) ←

**(**

*T*_{n}_{}= a + (*n*– 1)*d**T*

_{7}= 12

a + 6

*d*= 12 —– (2)(2) – (1),

4

*d =*– 8*d =*– 2

Substitute

*d =*– 2 into (1),*a*+ 2 (– 2) = 20

*a*= 24

*T*

_{20}=

*a*+ 19

*d*

*T*

_{20}= 24 + 19 (– 2)

*T*

_{20}**= –4**

(b)

(b)

*T*

_{n}= –34

a + (

*n*– 1)

*d*= –34

24 + (

*n*– 1) (–2) = –34(

*n*– 1) (–2) = –58*n*– 1 = 29

*n***= 30**

**The volume of water in a tank is 75 litres on the first day. Subsequently, 15 litres of water is added to the tank everyday.**

Example 3:

Example 3:

Calculate the volume, in litres, of water in the tank at the end of the 12th day.

Solution:Solution:

Volume of water on the first day = 75

*l*Volume of water on the second day = 75 + 15 = 90

*l*Volume of water on the third day = 90 + 15 = 105

*l*75, 90, 105, …..

AP,

*a*= 75,*d*= 90 – 75 = 15Volume of water on the 12

^{th}day,*T*

_{12}=

*a*+ 11

*d*

*T*

_{12}= 75 + 11 (15)

*T*

_{12}**= 240**

*l*

Example 4:

Example 4:

The first three terms of an arithmetic progression are 72, 65 and 58.

The

*n*th term of this progression is negative.Find the least value of

*n*.

*Solution:*72, 65, 58

AP,

*a*= 72,*d*= 65 – 72 = –7The nth term is negative,

*T*

_{n }< 0

*a*+ (

*n*– 1)

*d*< 0

72 + (

*n*– 1)*(–7) < 0*(

*n*– 1)*(–7) < –72**n*– 1 > –72

**/**–7

*n*– 1 > 10.28

*n*> 11.28

*n*must be integer,

*n*= 12, 13, 14, ….

**Therefore, the least value of**

*n*= 12.