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**1.** Find the value of ‘p’ if the numbers x, 2x + p, 3x + p are three successive terms of the AP.

**2.** Find p and q such that: 2p, 2p + q, p + 4q, 35 are in AP

**3.** Find a, b and c such that the following numbers are in A.P.

a, 7, b, 23, c

**Hint:**

7 – a = b – 7 ⇒ a + b = 14

23 – b = b – 7 ⇒ 2b = 30 ⇒ b = 15

23 – b = c – 23 ⇒ c + b = 4 6 ⇒ c = 46 – b

= 46 – 15

=31

And a = 14 – b = 14 – 15 = – 1**4.** Determine k so that k^{2} + 4k + 8, 2k^{2} + 3k + 6, 3k^{2} + 4k + 4 are three consecutive terms of an AP.

**5. ** are three consecutive terms of an AP, find the value of a.

**6.** For what value of p, are (2p – 1), 7 and three consecutive terms of an AP?

**7. **If (x + 2), 2x, (2x + 4) are three consecutive terms of an AP, find the value of x.

**8.** For what value of p are (2p – 1), 13 and (5p – 10) are three consecutive terms of an A.P.?

**9.** Find the 10th term from the end of the A.P. 4, 9, 14, ... 254.

**10.** Find the 6th term of the AP 54, 51, 48...

**11.** Find the 8th term from the end of the AP : 7, 10, 13, ..., 184.

**12.** Find the 16th term of the AP 3, 5, 7, 9, 11, ...

**13.** Find the 12th term of the AP:

14, 9, 4, –1, –6, ...

**14.** Find the middle term of the AP :

20, 16, ..., –180

**15.** Find the 6th term from the end of the A.P.

17, 14, 11, ..., (–40)

**16.** Find the middle term of the AP :

10, 7, 4, ..., (–62)

**17. **Which term of the AP : 24, 21, 18, 13, ... is the first negative term?**Hint:** The first negative term will be the term immediately less than 0. i.e. Tn < 0.

⇒ [a + (n – 1)d] < 0

Here, a = 24

d = (21 - 24) = -3

⇒ 3n > 27

⇒ n > 9

∴ n = 10**18.** The 6th term of an AP is –10 and its 10th term is –26. Determine the 15th term of the A.P.

**19.** For what value of n are the nth terms of the following two APs the same: 13, 19, 25, ... and 69, 68, 67, ....

**20.** The 8th term of an AP is zero. Prove that its 38th term is triple its 18th term.**Hint:**

T_{8} =0 ⇒ a + 7d = 0 ⇒ a = –7d

T_{38} = a + 37d = –7d + 37d = 30d

Also, T_{18} = a + 17d = –7d + 17d = 10d

30d = 3 × (10d) ⇒ T_{38} = 3 × T_{18}

**21.** For what value of n, the nth terms of the following two AP’s are equal?

23, 25, 27, 29, ... and –17, –10, –3, 4, ...

**22. **Which term of the AP : 5, 15, 25, ... will be 130 more than 31st term?**Hint:** Let an be the required term

i.e. an be 130 more than a_{31}

⇒ a_{n} – a_{31} = 130

**23.** Which term of the AP : 3, 15, 27, 39, ... will be 130 120 more than its 64th term?

**24.** The 9th term of an AP is 499 and its 499th term is 9. Which of its term is equal to zero.

**25.** Determine A.P. whose fourth term is 18 and the difference of the ninth term from fifteenth term is 30.

**26.** How many natural numbers are there between 200 and 500 which are divisible by 7?**Hint:** 200 ... 203 ... 497 ... 500

← Divisible by 7 →

∴ a = 203, d = 7 and a_{n} = 497

⇒ a + (n – 1) d = a_{n} ⇒ 203 + (n – 1) × 7 = 497**27. **How many multiples of 7 are there between 100 and 300?

**28.** Find the value of the middle term of the following A.P. : –11, –7, –3, ..., 49.

**29.** Find the value of the middle term of the following A.P. : –6, –2, 2, ..., 58.

**30.** How many two digit numbers are divisible by 3?**Hint:** Here, a = 12, d = 3 and an = 99

**31.** If the 9th term of an AP is zero, show that 29th term is double the 19th term.**Hint:**

⇒ 20d = 20d

⇒ a_{29} = a_{19}

**32.** If in an AP, the sum of its first ten terms is –80 and the sum of its next ten terms is –280. Find the AP

**33.** If in an A.P. a_{n} = 20 and S_{n} = 399 then find ‘n’**Hint:** a_{n} = a + (n – 1)d ⇒ (n – 1)d = 19

**34.** Find the sum of all natural numbers from 1 to 100.

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

**36.** How many terms of A.P. a, 17, 25, ... must be taken to get a sum of 450?

**37. **Find the sum of first hundred even natural numbers which are multiples of 5.

**38.** Find the sum of the first 30 positive integers divisible by 6.

**39.** Find the sum of those integers from 1 to 500 which are multiples of 2 or 5.**Hint:** Multiples of 2 are : 2, 4, 6, 8, 10, 12, 14, 16, ..., 500.

Multiples of 5 are : 5, 10, 15, 20, 25, 30, ..., 500.

Multiples of 2 as well as 5 : 10, 20, 30, 40, ..., 500. ∴ The required sum

= [Sum of multiplies of 2] +[S um of multiples of 5]-[ Multiples of 2 as we]

**40. **If the nth term of an A.P. is 2n + 1, find Sn of the A.P.

**41.** An A.P. consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the A.P.

**42. **If S_{n} denotes the sum of n-terms of A.P. whose common differences is d and first term is a find: S_{n} – 2S_{n–1} + S_{n–2}

Hint: a_{n} = S_{n} – S_{n–1}

**43.** If the ratio of 11th term to 18th term of an A.P. is 2 : 3. Find the ratio of the 5th term to the 21st term and also the ratio of the sum of the first five terms to the sum of first 21 terms.

**44.** If in an A.P. the first term is 2, the last term is 29 and sum of the terms is 155. Find the common difference of the A.P.

**45.** The sum of n terms of an A.P. is Find the 20th term.

**46.** If Sn denotes the sum of first n terms of an A.P., prove that

S_{30} = 3(S_{20} – S_{10})

**47.** The 4^{th} term of an A.P. is zero. Prove that the 25^{th} term of the A.P. is three times its 11^{th} term.

**48. **Find the 9^{th} term from the end (towards the first term) of A.P. 5, 9, 13, .........185.

**49.** How many terms of the A.P. 18, 16, 14, ......... be taken so that their sum is zero?

**ANSWERS**

**1.** p = 0

**2.** p = 10, q = 5

**3.** a = –1, b = 15, c = 31

**4.** k = 0

5. a = 8/5

**6.** p = 2

**7.** x = 6

**8.** p = 5

**9.** 209

**10.** 69

**11.** 163

**12****.** 33

**13.** –41

**14.** –80

**15.** –25

**16.** –26

**17.** n = 10

**18. **–46

**19.** n = 9

**21.** n = 9

**22.** 44th

**23. **74th

**24.** 508

**25.** 3, 8, 13, 18, ...

**26.** 43

**27.** 28

**28.** 17; 21

**29**. 26

**30.** 30

**32.** 1, –1, –3, –5, –7...

**33.** 38

**34.** 5050

**35. **12, 510

**36.** 10

**37.** 50500

**38.** 2790

**39.** 27250

**40.** n(n + 2)

**41.** 3, 7, 11, 15, ...

**42.** d

**43.** 1 : 3; 5 : 49

**44.** d = 3

**45.** 99

**48. **153

**49.** n = 19

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