CHAPTER – 5
ARITHMETIC PROGRESSION
nth Term of an ARITHMETIC PROGRESSION ( AP )
nth term an of the AP with first term a and common difference d is given by
an = a + (n – 1) d.
IMPORTANT QUESTIONS
Find the 15th term of the 21, 24, 27, . . .
Solution: Here, a = 21, d = 24 – 21 = 3
We know that an = a + (n – 1)d
So, a15 = a + 14d = 21 + 14(3) = 21 + 42 = 63
Which term of the AP : 3, 9, 15, 21, . . . , is 99?
Solution: Here, a = 3, d = 9 – 3 = 6
We know that an = a + (n – 1)d
Let an = 99 a + (n – 1)d = 99
3 + (n – 1)6 = 99 (n – 1)6 = 99 – 3 = 96
96
n 1 16 n = 16 + 1 = 17
6
Hence, 17th term of the given AP is 99
Determine the AP whose 3rd term is 5 and the 7th term is 9.
Solution: We have a3 = a + (3 – 1) d = a + 2d = 5 ............... (1)
and a7 = a + (7 – 1) d = a + 6d = 9 ........................ (2)
Solving the pair of linear equations (1) and (2), we get a = 3, d = 1
Hence, the required AP is 3, 4, 5, 6, 7, . . .
Questions for practice
1. Find the 10th term of the AP : 2, 7, 12, . . .
2. Which term of the AP : 21, 18, 15, . . . is – 81?
3. Which term of the AP : 3, 8, 13, 18, . . . ,is 78?
4. How many two-digit numbers are divisible by 3?
5. How many three-digit numbers are divisible by 7?
6. How many multiples of 4 lie between 10 and 250?
7. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
8. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
9. If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
10. Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?
11. Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
12. The sum of 4th term and 8th term of an AP is 24 and the sum of 6th and 10th terms is 44. Find
the AP.
13. The sum of 5th term and 9th term of an AP is 72 and the sum of 7th and 12th terms is 97. Find
the AP.
14. If the numbers n – 2, 4n – 1 and 5n + 2 are in AP, find the value of n.
15. Find the value of the middle most term (s) of the AP : –11, –7, –3,..., 49.
16. The sum of the first three terms of an AP is 33. If the product of the first and the third term
exceeds the second term by 29, find the AP.
17. The sum of the 5th and the 7th terms of an AP is 52 and the 10th term is 46. Find the AP.
, 18. Find the 20th term of the AP whose 7th term is 24 less than the 11th term, first term being 12.
19. If the 9th term of an AP is zero, prove that its 29th term is twice its 19th term.
20. The 26th, 11th and the last term of an AP are 0, 3 and –1/5 , respectively. Find the common
difference and the number of terms.
21. Find whether 55 is a term of the AP: 7, 10, 13,--- or not. If yes, find which term it is.
22. Determine k so that k2+ 4k + 8, 2k2 + 3k + 6, 3k2 + 4k + 4 are three consecutive terms of an AP.
23. Split 207 into three parts such that these are in AP and the product of the two smaller parts is
4623.
24. The angles of a triangle are in AP. The greatest angle is twice the least. Find all the angles of the
triangle.
25. If the nth terms of the two APs: 9, 7, 5, ... and 24, 21, 18,... are the same, find the value of n.
Also find that term.
26. If sum of the 3rd and the 8th terms of an AP is 7 and the sum of the 7th and the 14th terms is –3,
find the 10th term.
27. Which term of the AP: 53, 48, 43,... is the first negative term?
28. 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 form an AP? If so, find the interest at the end of 30 years making
use of this fact.
29. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so
on. There are 5 rose plants in the last row. How many rows are there in the flower bed?
nth Term from the end of an ARITHMETIC PROGRESSION ( AP )
Let the last term of an AP be ‘l’ and the common difference of an AP is ‘d’ then the nth term from
the end of an AP is given by
ln = l – (n – 1) d.
IMPORTANT QUESTIONS
Find the 11th term from the last term (towards the first term) of the AP : 10, 7, 4, . . ., – 62.
Solution : Here, a = 10, d = 7 – 10 = – 3, l = – 62,
We know that nth term from the last is given by ln = l – (n – 1) d.
l11 = l – 10d = – 62 – 10(– 3) = – 62 + 30 = – 32
Questions for practice
1. Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.
2. Find the 10th term from the last term of the AP : 4, 9 , 14, . . ., 254.
3. Find the 6th term from the end of the AP 17, 14, 11, …… (–40).
4. Find the 8th term from the end of the AP 7, 10, 13, …… 184.
5. Find the 10th term from the last term of the AP : 8, 10, 12, . . ., 126.
6. Find the 12th term from the end of the AP: –2, –4, –6,..., –100.
Sum of First n Terms of an ARITHMETIC PROGRESSION ( AP )
The sum of the first n terms of an AP is given by
n
S n [2a (n 1)d ]
2
where a = first term, d = common difference and n = number of terms.
or
n
S n [a l ]
2
where l = last term
ARITHMETIC PROGRESSION
nth Term of an ARITHMETIC PROGRESSION ( AP )
nth term an of the AP with first term a and common difference d is given by
an = a + (n – 1) d.
IMPORTANT QUESTIONS
Find the 15th term of the 21, 24, 27, . . .
Solution: Here, a = 21, d = 24 – 21 = 3
We know that an = a + (n – 1)d
So, a15 = a + 14d = 21 + 14(3) = 21 + 42 = 63
Which term of the AP : 3, 9, 15, 21, . . . , is 99?
Solution: Here, a = 3, d = 9 – 3 = 6
We know that an = a + (n – 1)d
Let an = 99 a + (n – 1)d = 99
3 + (n – 1)6 = 99 (n – 1)6 = 99 – 3 = 96
96
n 1 16 n = 16 + 1 = 17
6
Hence, 17th term of the given AP is 99
Determine the AP whose 3rd term is 5 and the 7th term is 9.
Solution: We have a3 = a + (3 – 1) d = a + 2d = 5 ............... (1)
and a7 = a + (7 – 1) d = a + 6d = 9 ........................ (2)
Solving the pair of linear equations (1) and (2), we get a = 3, d = 1
Hence, the required AP is 3, 4, 5, 6, 7, . . .
Questions for practice
1. Find the 10th term of the AP : 2, 7, 12, . . .
2. Which term of the AP : 21, 18, 15, . . . is – 81?
3. Which term of the AP : 3, 8, 13, 18, . . . ,is 78?
4. How many two-digit numbers are divisible by 3?
5. How many three-digit numbers are divisible by 7?
6. How many multiples of 4 lie between 10 and 250?
7. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
8. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
9. If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
10. Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?
11. Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
12. The sum of 4th term and 8th term of an AP is 24 and the sum of 6th and 10th terms is 44. Find
the AP.
13. The sum of 5th term and 9th term of an AP is 72 and the sum of 7th and 12th terms is 97. Find
the AP.
14. If the numbers n – 2, 4n – 1 and 5n + 2 are in AP, find the value of n.
15. Find the value of the middle most term (s) of the AP : –11, –7, –3,..., 49.
16. The sum of the first three terms of an AP is 33. If the product of the first and the third term
exceeds the second term by 29, find the AP.
17. The sum of the 5th and the 7th terms of an AP is 52 and the 10th term is 46. Find the AP.
, 18. Find the 20th term of the AP whose 7th term is 24 less than the 11th term, first term being 12.
19. If the 9th term of an AP is zero, prove that its 29th term is twice its 19th term.
20. The 26th, 11th and the last term of an AP are 0, 3 and –1/5 , respectively. Find the common
difference and the number of terms.
21. Find whether 55 is a term of the AP: 7, 10, 13,--- or not. If yes, find which term it is.
22. Determine k so that k2+ 4k + 8, 2k2 + 3k + 6, 3k2 + 4k + 4 are three consecutive terms of an AP.
23. Split 207 into three parts such that these are in AP and the product of the two smaller parts is
4623.
24. The angles of a triangle are in AP. The greatest angle is twice the least. Find all the angles of the
triangle.
25. If the nth terms of the two APs: 9, 7, 5, ... and 24, 21, 18,... are the same, find the value of n.
Also find that term.
26. If sum of the 3rd and the 8th terms of an AP is 7 and the sum of the 7th and the 14th terms is –3,
find the 10th term.
27. Which term of the AP: 53, 48, 43,... is the first negative term?
28. 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 form an AP? If so, find the interest at the end of 30 years making
use of this fact.
29. In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so
on. There are 5 rose plants in the last row. How many rows are there in the flower bed?
nth Term from the end of an ARITHMETIC PROGRESSION ( AP )
Let the last term of an AP be ‘l’ and the common difference of an AP is ‘d’ then the nth term from
the end of an AP is given by
ln = l – (n – 1) d.
IMPORTANT QUESTIONS
Find the 11th term from the last term (towards the first term) of the AP : 10, 7, 4, . . ., – 62.
Solution : Here, a = 10, d = 7 – 10 = – 3, l = – 62,
We know that nth term from the last is given by ln = l – (n – 1) d.
l11 = l – 10d = – 62 – 10(– 3) = – 62 + 30 = – 32
Questions for practice
1. Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.
2. Find the 10th term from the last term of the AP : 4, 9 , 14, . . ., 254.
3. Find the 6th term from the end of the AP 17, 14, 11, …… (–40).
4. Find the 8th term from the end of the AP 7, 10, 13, …… 184.
5. Find the 10th term from the last term of the AP : 8, 10, 12, . . ., 126.
6. Find the 12th term from the end of the AP: –2, –4, –6,..., –100.
Sum of First n Terms of an ARITHMETIC PROGRESSION ( AP )
The sum of the first n terms of an AP is given by
n
S n [2a (n 1)d ]
2
where a = first term, d = common difference and n = number of terms.
or
n
S n [a l ]
2
where l = last term
