# ARITHMETIC PROGRESSION

ARITHMETIC PROGRESSION (AP) An arithmetic progression is the list of the numbers in which each term is obtained by adding a fixed number to the preceeding term except the first term. This fixed number is called the common difference (d) of the AP. It can be positive ,negative or zero.
General form of an AP :
$a,a+d,a+2d,a+3d,.....$

#### nth term of an AP from the beginning (General Term) :

General term ( or nth term ) from the beginning of an AP is



#### Sum of the first nth terms of an AP :

The sum of the first n terms of an AP is given by

## ARITHMETIC PROGRESSION IMPORTANT QUESTION

1. Write the first three terms of the AP when a and d are as given below :
a = -5, d = -3
2. If k + 1, 3k and 4k + 2 be any three consecutive terms of an AP, find the value of k.

3. Determine the AP whose third term is 16 and the seventh term exceeds the fifth term by 12.

4. 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.

5. If the 9th term of an AP is zero, Prove that its 29th term is twice its 19th term. (CBSE 09)

6. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.       (CBSE 08, 09)

7. Find the value of the middle term of the AP : -6, -2, 2, ..........., 58.  (CBSE 11)

8. Which term of the sequence 114, 109, 104 .... is the first negative term?

9. The sum of three numbers in an AP is 27 and their product is 405. Find the numbers.

10. Find four numbers in AP whose sum is 20 and the sum of whose squares is 120.

11. The sum of the four consecutive numbers in AP is 32 and the ratio of the product of the first and the last terms to the product of the two middle terms is 7 : 15. Find the numbers.     (HOT)

12. If m times the mth term of an AP is equal to n times its nth term, them show that (m + n )th of the AP is zero.    (CBSE 2004, 08)

13. If  (an + 1 +  bn + 1) / (an  +  bn)  is the arithmetic mean between a and b, find the value of n.
(HOT)

14. The Sum of n terms of an AP is 3n2 + 5n . Find the AP. Hence , Find Its 16th term. (CBSE 08)

15. If the sum of the first 4 terms of an AP is 40 and that of the first 14 terms is 280, find the sum of its first n  terms.              (CBSE 2011)

16. Show that the sum of an AP whose first term is a, the second term b and the last term c, is given by       S = (a + c).( b + c - 2a) / 2(b - a)             (HOT)

17. The sum of the first seven terms of an AP is 182. if the 4th and 7th terms are in the ratio 1 : 5, find the AP.   ( CBSE 2014)

18. The first and last term of an AP are 4 and 81 respectively. If the common difference is 7, how many terms are there in the AP and what is their sum?

19. Find the sum of all natural numbers lying between 100 and 500 which are divisible by 8.

20. Jaspal Singh repays his loan of Rs118000 by paying every month starting with the first instalment of Rs1000. If he increases the instalment by Rs100 every month, what amount will be paid by him in the 30th instalment? What loan does he still have to pay after the 30th instalment ?

21. The students of a school decided to beautify the school on the annual day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books ? What is the maximum Distance she travelled carrying a flag?

22. If the pth term of an AP be 1 / q and the qth term be 1 / p, then prove that the sum of the first pq terms must be (pq + 1) / 2.

23. A circle is completely divided into n sectors in such a way that the angles of the sectors are in arithmetic progression. If the smallest of these angles is 8॰ and the largest 72॰ , calculate n and the angle in the fourth sector.

24. Find the sum of all two digit numbers greater than 50 which when divided  by 7 leave a remainder of 4.

25. If Sdenotes the sum of Sn terms of an AP, prove that  S12 =  3(S8 -S4).

## ARITHMETIC PROGRESSION - ANSWERS OF IMPORTANT QUESTIONS

1. -5, -8, -11
2.  3
3.  4, 10, 16, 22,...............
4.  d= -1/5  n = 27
6.  -13, -8, -3
7.  middle term is 9th term,    value = 26
8.  n = 24
9.  (3, 9, 15) or (15, 9, 3)
10. (2, 4, 6, 8) or (8, 6, 4, 2)
11. (2, 6, 10, 14) or (14, 10, 6,2)
13. n =0
14. (8, 14, 20) & 98
15. n(n + 6)
17. 17, 20, 23, .............
18.  n = 12, S = 510
19. 15000
20. Rs 3900, Rs 44500
21. 26 m.
23. 32॰
24. 518