# EXERCISE 19.4

**QUESTION 1**

Find the sum of the following arithmetic progressions:

(i) to 10 terms

Sol :

We have :

a = 50

d = (46 -50) = -4

n = 10

= 320

(ii) to 12 terms

Sol :

We have :

a = 1

d = (3 – 1) = 2

n = 12

= 144

(iii) to 25 terms

Sol :

We have :

a = 3

n = 25

= 525

(iv) to 12 terms

Sol :

We have :

a = 41

d = (36 – 41) = -5

n = 12

= 162

(v) to 22 terms

Sol :

We have :

First term = a + b

d = (a – b – a – b) = -2b

n = 22

(vi) to terms

Sol :

We have :

(vii) to terms

Sol :

We have :

**QUESTION 2**

Find the sum of the following series:

(i)

Sol :

Here, the series is an A.P. where we have the following:

a = 2

d = (5 – 2) = 3

= 5412

(ii)

Sol :

Here, the series is an A.P. where we have the following:

a = 101

d = (99 – 101) = -2

= 2072

(iii)

Sol :

Here, the series is an A.P. where we have the following:

= 2*ab*

**QUESTION 3**

Find the sum of first *n* natural numbers.

Sol :

The first natural numbers are:

a = 1 , d = 1

Total terms = *n*

**QUESTION 4**

Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5

Sol :

We have to find the sum of all the natural numbers from 1 to 100 , that are divisible by 2 or 5 .

**S = A + B – C**

**S** = Requires sum

**A** = sum of natural numbers between 1 to 100 divisible by 2

**B** = sum of all the natural number between 1 to 100 divisible by 5

**C** = sum of all natural number between 1 to 100 , which is divisible by 2 and 5 both (just to remove those number which is already present in *A* and *B* i.e 10 )

A

where *a = 2 , *

B

where *a = 5 *,

C

where *a = 10 *,

S =

= 2500 + 1000 -500

= 3000

**QUESTION 5**

Find the sum of first *n* odd natural numbers.

Sol :

The first *n* odd natural numbers are: 1 , 3 , 5 , 7 …..

a = 1 , d = 2

Total terms = n

**QUESTION 6**

Find the sum of all odd numbers between 100 and 200

Sol :

All the odd numbers between 100 and 200 are:

101 , 103 , …… 199

Here , we have

a = 101

d = 2

**QUESTION 7**

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667

Sol :

The odd integers between 1 and 1000 that are divisible by 3 are:

3 , 9 , 15 , 21 ….. 999

Here , we have

a = 3 , d = 6

=83667

Hence , proved

**QUESTION 8**

Find the sum of all integers between 84 and 719, which are multiples of 5

Sol :

The integers between 84 and which are multiples of 5 are:

85 , 90 . . .715

Here , we have

a = 5 , d = 5

=50800

**QUESTION 9**

**QUESTION 10**

**QUESTION 11**

**QUESTION 12**

Find the sum of the series :

3 + 5 + 7 + 6 + 9 + 13 + 17 + …… to 3*n *terms .

Sol :

The given sequence can be rewritten as 3 + 6 + 9… to *n * terms + 5 + 9 + 13 +. … to *n * terms + 7 + 12 + 17+ … to *n *terms

Clearly , all these sequence forms an A.P. having *n* terms with first terms 3 , 5 , 7 and common difference 3 , 4 , 5

Hence , required sum

**QUESTION 13**

Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the

remainder 7

Sol :

The sum of all those integers between 100 and 800 each of which on division by 16 leaves the

remainder 7 are:

**QUESTION 14**

**QUESTION 15**

**QUESTION 16**

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**QUESTION 19**

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**QUESTION 26**

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**QUESTION 34**