## nth Term of an Arithmetic Progression

Let …….. be an A.P., with first term as *a*, and common difference as *d*.

First term is = *a* (i)

Second term = *a + d *(ii)

=

Third term = (iii)

= [from (i)]

=

or =

Fourth term

or = [from (iii)]

=

=

\ nth term *a _{n}*

*Note: The nth term of the A.P. with first term a & common difference d is given by *

is also called as general term of an A.P.

If there are P terms in the A.P. then *a _{p}* represents the last term which can also be denoted by

*l*.

## Problems based on nth TERM OF AN Arithmetic Progression

### Problem:

#### Find the 18^{th} term and *n*^{th} term for the sequence 7, 4, 1, –2, –5.

### Solution:

Here *a* = 7

and *d* =

= 4 – 7 = – 3

n = 18

= 7 + 17 ´ –3

= 7 – 51

= –44

= 7 + (*n *– 1) (–3)

= 7 – 3n + 3

= 10 – 3n

### Problem:

#### Which term of the A.P. 7, 12, 17, ….. is 87?

### Solution:

*a* = 7

= 12 – 7 = 5

As

87 = 7 + (*n *– 1) ´ 5

17 = *n
*

\ 17^{th} term of give A.P. is 87

### Problem:

#### How many terms are there in A.P. 7, 13, 19, …….., 205.

### Solution:

*a* = 7

= 13 – 7 = 6

205 – 7 = 6*n*– 6

198 + 6 = 6*n
*

or 34 = *n*

\ Given A.P. has 34 terms

### Problem:

#### Check whether –150 is term of the A.P. 11, 8, 5, 2, …….?

### Solution:

*a* = 11

= 8 – 11 = -3

let [Assume]

As number of term can’t be in fraction, \ –150 is not a term of the given A.P.

### Problem:

#### In the following A.P. find the missing terms in the boxes.

#### (i) 5, , , (ii) –4, , , , , 6

### Solution:

(i) Here *a* = 5

& or

Now,

\

\

Hence

(ii) Here

As

6 = –4 + 5*d*

*d *= 2

\

### Problem:

#### For what value of *n*, the *n*th terms of A.P’s 63, 65, 67, ……… and 3, 10, 17, …… are equal.

### Solution:

First sequence is 63, 65, 67, ……..

*a* = 63

*d*_{1} = 65 – 63 = 2

= 63 + 2*n*–2

= 61 + 2*n
*

Second sequence is 3, 10, 17, ………..

\

\

=

= 7*n*–4

According to question

61 + 2*n* = 7*n*– 4

61 + 4 = 7*n*– 2*n*

65 = 5*n*

\*n* = 13

### Problem:

#### Two A.P.s have the same common difference. If the difference between their 100^{th} terms is 100, what is the difference between their 1000^{th} terms?

### Solution:

Let the common difference of two A.P.s be *d
*

* *Then, their 100^{th} terms will be

According to question

i.e. …(i)

Now, difference between their 1000^{th} terms

[By equation (i)]

### Problem:

#### Which term of the arithmetic progression 3, 10, 17, …. will be 84 more than its 13^{th} term?

### Solution:

The give A.P. is 3, 10, 17, ……

Here, *a* (first term) = 3

*d* (common difference)= 10 – 3 = 7

= 3 + 12 ´ 7 = 3 + 84 = 87

Let *n*^{th} term be 84 more than the 13^{th} term of the given A.P.

So, we get

\ The 25^{th} term of the given A.P. will be 84 more than its 13^{th} term.

### Problem:

#### Find the 31^{st} term of an A.P. whose 11^{th} term is 38 and the 16^{th} term is 73.

### Solution:

Let *a* be the 1^{st} term and *d* the common difference.

Here …(i)

…(ii)

Subtracting (ii) and (i), we get

or

Putting *d* = 7 in (i), we get

\

Hence, 31^{st} term is 178.

### Problem:

#### How many three digit numbers are divisible by 7?

### Solution:

Three digit numbers which are divisible by 7 are 105, 112, 119, …, 994.

Here,

\

Þ

Þ

Þ *n *– 1 = 127

*n* = 127 + 1 = 128.

\ There are 128 three digit numbers which are divisible by 7

### Problem:

#### A sum of Rs 1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these interests from an AP? If so, find the interest at the end of 30 years.

### Solution:

We know that the formula to calculate simple interest is given by

Simple Interest =

So, the interest at the end of the 1^{st} year = Rs = Rs 80

The interest at the end of the 2^{nd} year = Rs = Rs 160

The interest at the end of the 3^{rd} year = Rs = Rs 240

Similarly, we can obtain the interest at the end of the 4^{th} year, 5^{th} year, and so on.

So, the interest (in Rs) at the end of the 1^{st}, 2^{nd}, 3^{rd}, ….. years, respectively are 80, 160, 240, …

It is an AP as the difference between the consecutive terns in the list is 80, i.e., *d* = 80. Also, *a* = 80.

So, to find the interest at the end of 30 years, we shall find .

Now,

So, the interest at the end of 30 years will be Rs 2400.

## TO FIND nth TERM FROM THE END OF AN A.P.

Consider the following A.P. ,

where l is the last term

last term = – (1 – 1)*d
*

2^{nd} last term –

*d* = – (2 – 1) *d*

3^{rd} last term – 2*d* = – (3 – 1)*d*

……………………………

……………………………

Note: *nth term from the end = **– (n – 1) d*

### Problem:

#### Find the 5^{th} term from the end of the AP, 17, 14, 11, ….., –40

### Solution:

**1 ^{st} method**

Using

5^{th} term from the end will be

= –28

**2 ^{nd} method
**

Sequence can be written as –40, –37, ….11, 14, 17

\

=–37 + 40

= 3

*n* = 5

Using

=

= –40 + 12

= –28

### Problem:

#### The sum of the 4^{th} and 8^{th} terms of an A.P. is 24 and sum of 6^{th} & 10^{th} terms is 44. Find the first three term of the AP.

### Solution:

Using 1^{st} condition

\

or …(i)

Using 2^{nd} condition

or

or …(ii)

Subtracting equation (i) from (ii) we have

or *d* = 5

Putting value of *d* in equations (i)

\

### Problem:

#### If the *p*th term of an A.P. is *q* and the *q*th term is *p*, prove that its nth term is

### Solution:

Let *a* be the first term and *d* be the common difference of the given A.P.

Then, pth term = *q* Þ …(i)

qth term = *p* Þ …(ii)

Subtracting equation (ii) from equation (i), we get

…(iii)

Putting in equation (i), we get

\ *n*th term

[from equation (iii)]

### Problem:

#### If *m* times the *m*th term of an A.P. is equal to *n* times its *n*th term, show that the *(m + n)* term of the A.P. is zero.

### Solution:

Let *a* be the first term and *d* be the common difference of the given A.P.

According to question, *m* times the *m*th term = *n* times the *n*th term

Þ

Þ

Þ

Þ

Þ

Þ

Þ

Þ

Þ []

Þ

\ th term of A.P. is zero

## CONDITION FOR TERMS TO BE IN A.P.

If three numbers *a, b, c,* in order are in A.P. Then,

common difference =

Þ

Þ

*Note: a, b, c are in A.P. iff *

### Problem:

#### If are in A.P., find the value of *x*.

### Solution:

Since, are in A.P.

\

Þ

Þ

Þ *x* = 6

### Problem:

#### If the numbers *a*, *b*, *c*, *d*, *e* form an A.P., then find the value of .

### Solution:

Let *D* be the common difference of the given A.P. Then,

and

\

Þ

Þ

Þ