7. Chapter 7. Arithmetic Progressions NIOS Class 10 Mathematics Textbook Solutions
Chapter 7. Arithmetic Progressions NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 7. ARITHMETIC PROGRESSIONS
CHECK YOUR PROGRESS 7.1
Which of the following are AP? If they are in AP, find their first terms and common differences:
1. –5, –1, 3, 7, 11, ....
2. 6, 7, 8, 9, 10, ...
3. 1, 4, 6, 7, 6, 4, ....
4. –6, – 3, 0, 3, 6, 9, ....
Solution: 1. –5, –1, 3, 7, 11, ....
Since, 
, 
, 
,, 
It is an arithmetic progression (AP).
First term = – 5 and common difference = 4 .
2. 6, 7, 8, 9, 10, ...........
Since, 
It is an arithmetic progression (AP).
First term = 6 and common difference = 1 .
3. 1, 4, 6, 7, 6, 4, .........
Since, 
It is not an arithmetic progression (AP).
4. –6, – 3, 0, 3, 6, 9, ....
Since, 
It is an arithmetic progression (AP).
First term = – 6 and common difference = 3 .
CHECK YOUR PROGRESS 7.2
1. The first term of an AP is 4 and common difference is – 3, find its 12th term.
Solution: Here, 
We know that, 
2. The first term of an AP is 2 and 9th term is 26, find the common difference.
Solution: Here, 
We have,
3. The 12th term of an AP is – 28 and 18th term is – 46. Find its first term and common difference.
Solution: Let a and d be the first term and common difference of an A.P. respectively .
We know that,
A/Q, 
(i)
And 
[ From (i)]
Putting 
in equation (i) , we get 
Therefore, the first term = 5 and the common difference = – 3
4. Which term of the AP 5, 2, –1, .... is – 22?
Solution: Here, 
We have,
5. If the pth , qth and rth terms of an AP are x, y and z respectively, prove that: 
Solution: Let a and d be the first term and common difference of an A.P. respectively .
We know that,
A/Q, 
(i)
(ii)
And 
(iii)
LHS : 
RHS .
LHS = RHS Proved.
CHECK YOUR PROGRESS 7.3
1. Find the sum of first 15 terms of the following APs:
(i) 11, 6, 1, – 4, –9 ...........
(ii) 7, 12, 17, 22, 27 .............
Solution: (i) Here, 
We know that, 
(ii) Here, 
We know that, 
2. How many terms of the AP 25, 28, 31, 34, .... are needed to give the sum 1070 ?
Solution: Here, 
We have, 
or 
Therefore, the number of term is 20 .
3. Find the following sum: 1 + 4 + 7 + 10 + .... +118
Solution: Here, 
We have,
We know that , 
4. Find the sum of all natural numbers upto 100 which are divisible by 3.
Solution: The list of natural number divisible by 3 are : 3 , 6 , 9 , 12 , ……….99 .
Here, 
We have,
Again, 
5. The sum of any three consecutive terms of an AP is 21 and their product is 231. Find the three terms of the AP.
Solution: Let, 
are the three numbers.
A/Q, 
and 
For 
, then 
For 
, then 
Therefore, the three terms of an A.P are : 3 , 7 and 11 or 11 , 7 and 3 .
6. Of the 
, determine the ones that are missing for each of the following arithmetic progression
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
We know that,
or 
We know that,
(ii) 
We have, 
(i)
And 
From (i) we get , 
(iii) 
We have,
And
(iv) 
We have, 
and 
TERMINAL EXERCISE
1. Which of the following patterns are arithmetic progressions?
(i) 2, 5, 8, 12, 15, .... (ii) – 3, 0, 3, 6, 9 ..... (iii) 1, 2, 4, 8, 16, .....
Solution: (i) 2, 5, 8, 12, 15, ....
Since, 
It is not an arithmetic progression (AP) because the common differences are not equal.
(ii) – 3, 0, 3, 6, 9 .....
Since, 
It is an arithmetic progression (AP) because the common differences are equal.
(iii) 1, 2, 4, 8, 16, .....
Since, 
It is not an arithmetic progression (AP) because the common differences are not equal.
2. Write the nth term of each of the following arithmetic progressions:
(i) 5, 9, 13, 17, ....
(ii) – 7, – 11, – 15, – 19
Solution: (i) 5, 9, 13, 17, ....
Here , 
We have, 
(ii) – 7, – 11, – 15, – 19
Here , 
We have, 
3. The fourth term of an AP is equal to three times its first term and seventh term exceeds twice the third term by 1. Find the first term and common difference.
Solution: Let the first term be a and the common difference be d.
We know that,
A/Q, 
And 
Putting 
in (i) , we get
Therefore, First term = 3 and common difference = 2.
4. The 5th term of an AP is 23 and 12th term is 37. Find the first term and common difference.
Solution: Let a and d be the first term and common difference of an AP respectively.
A/Q, 
and 
Putting 
in (i) , we get 
Therefore, the first term and common difference are 15 and 2 respectively .
5. The angles of a triangle are in AP. If the smallest angle is one-third the largest angle ,find the angles of the triangle.
Solution: Let the angles be a − d , a and a + d .
A/Q, 
And 
Thus the angles are: 
i.e., 
6. Which term of AP
(i) 100, 95, 90, 85, ...., is – 25?
(ii) 
is..... 
?
Solution: (i) 100, 95, 90, 85, ...., is – 25?
Here, 
We have,
Therefore, the number of term is 26 .
(ii) is..... ?
Here, 
We have, 
Therefore, the number of term is 25 .
7. The nth term of an AP is given by 
. Show that it is an AP. Find its first term and common difference.
Solution: Given, 
Therefore, the first term is 
and common difference is b .
8. If 7 times the 7th term of an AP is equal to 11 times the 11th term, show that the 18th term is zero.
Solution: Let a and d be the first term and common difference of an AP respectively.
A/Q, 
Proved.
9. Each term of an AP whose first term is a and common difference is d, is doubled. Is the resulting pattern an AP? If so, find its first term and common difference.
Solution: Yes, if each term of an arithmetic progression (AP) is doubled, the resulting sequence is still an AP.
Let the original AP be: a,  a+d,  a+2d,  a+3d,  ……………
Doubling each term gives: 2a,  2(a+d),  2(a+2d),  2(a+3d),  ………..
i.e., 2a,  2a+2d,  2a+4d,  2a+6d,  …………….
The difference between consecutive terms = (2a+2d) – 2a=2d
Therefore, the first term = 2a and the common difference = 2d​
10. If 
are three consecutive terms of an AP, find k.
Solution: Since, are three consecutive terms of an AP .
Therefore, the value of k is 3 .
[Note: 
are three consecutive terms of an AP , then 
]
11. How many terms of the AP:
(i) 1, 4, 7, 10, .... are needed to get the sum 715?
(ii) –10, –7, –4, –1, ..... are needed to get the sum 104?
Solution: (i) 1, 4, 7, 10, .... are needed to get the sum 715?
Here , 
A/Q, 
or 
Therefore, the number of the term of an AP is 22.
(ii) –10, –7, –4, –1, ..... are needed to get the sum 104?
Here , 
A/Q, 
or 
Therefore, the number of the term of an AP is 13 .
12. Find the sum of first 100 odd natural numbers.
Solution: The first 100 odd natural numbers are: 1,3,5,7,………….,199
Here, 
We know that, 
13. In an AP, a = 2 and sum of the first five terms is one-fourth the sum of the next five terms. Show that its 20th term is –12.
[Hint: If AP is 
then 
]
In the next five terms, the first term is 
and last term is 
.
Solution: Here, 
Let the common difference be d .
We know that, 
A/Q, 
Now, 
14. If sum of first n terms of an AP is 
, find rth term of the A.P. [Hint 
]
Solution: Given, 
and
15. Find the sum of all 3-digit numbers which leave the remainder 1, when divided by 4.
[Hint: First term = 101, last term = 997]
Solution: Smallest 3-digit number is 100 and Largest 3-digit number: 999
The list of all 3-digit numbers which leave the remainder 1 when divided by 4 is : 100+1,104+1 ,108+1,…………..,996+1
i.e., 101,105,109,………….,997
Here, 
We have,
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