CBSE Important Questions

Section - A

Arithmetic Progression ( nth term)

If general term is given :

Q. 1. Write the first three terms of the sequence whose nth term is:

1. 3n – 5 / 6
2. n² / 3ⁿ.

Q. 2. What is the 17^th term of the sequence T_n = n (n + 3) / n + 1?

Q. 3. A sequence {a_n} is given by the formula an = 10 – 3n. Prove that it is an A.P.

Q. 4. A sequence {a_n} is given by an = n₂ – 1. Show that it is not an A.P.

Q. 5. If the nth term of an AP is (5n – 2), find its

1. fist term,
2. 19^th term, and
3. common difference.

Q. 6. If the nth term of a progression is (4n – 10), show that it is an AP. Find its

1. first term,
2. common difference, and
3. 16^th term.

If a, b and c are in A.P. then common difference are same
then b – a = c – b :

Q. 7. Determine k so that 2 / 3, k and 5 / 8 are the three consecutive terms of an A.P.

Q. 8. Determine k so that k + 2, 4k – 6 and 3k – 2 are the three consecutive terms of an A.P.

Q. 9. For what value of x are the terms 2x, (x + 10) and (3x + 2) in A.P.?

Q.10. Determine k so that 4k + 8, 2k² + 3k + 6, 3k² + 4k + 4 are three consecutive terms of an A.P.

Q.11. Show that (a – b)², (a² + b²) and (a + b) ² are in AP.

Q.12. Show that the sequence 6, 4, 2, 0, -2, -4, …., is an A.P. Write down its first term and the common difference.

General term , “an = a + (n-1)d”, where ‘a’ is first term, ‘d’ is common difference, ‘n’ is the position, ‘a’n is the value of the term at nth position :

Q.13. Find

1. nth term of 1/m, (1 + 2m)/m , (1 + 4m)/ m,…………..
2. find (n –1)th term of 17, 11, 5, -1,…………………
3. find nth term of the AP (a – b)^{2, (a2} + b²) and (a + b) ²………………

 

Q. 14. Find :-

1. The 8^th term of 117, 104, 91, 78,..
2. The 10^th term of 10.0, 10.5, 11.0, 11.5 ……
3. The 27^th and the nth terms of the sequence, 5, 2, -1, -7, ……
4. The 105^th term of the A.P. 4, 4 ½ , 5, 5 ½ , 6, ….
5. The 9^th term of 3/4 , 5/4, 7/4, 9/4………………..

Q.15. Find 12^th term of an AP with

1. first term 9 and common difference 10.
2. first term -20 and common difference 4.
3. first term 1/2 and common difference 1/12.

Q. 16. Which term of the A.P.

1. 5, 13, 21, ….is 181?
2. 5, 2, -1 ….is -22?
3. 2, 7, 12, 17,… is 87?
4. 21, 42, 63, 84, ….is 420?

Q.17. Find k, if given value of x is the kth term of the given AP.

1. 25,50,75,100,…….; x = 1000.
2. –1,-3,-5,-7,……….; x = -151
3. 5.5,11,16.5,22,……; x = 550
4. 1, 21/11, 31/11, 41/11, ……..; x = 171/11

Q. 18. Find the number of terms in the A.P.

1. 17, 14 ½ , 12 …. -38.
2. 7, 11, 15, …, 139?
3. first term = 5, the common difference = 3 and the last term = 83
4. 5 + 8 + 11 + 14 + ….. if the last term is 95.

Q. 19. which term of the AP 3,15,27,39,…. Will be the 132 more than its 54^th term.

Q. 20. Find

1. if 50 is a term of the A.P. 4, 7, 10, 13 …. or not. If yes, find which term it is ?
2. Is 51 a term of the AP 5, 8, 11, 14, …?

Q. 21. Which term of the sequence.

1. 17, 16 1/2, 14 3/5, …. is the first negative term?
2. 24, 21, 18, 15, … is the first negative term?

Q. 22. How many two-digit whole numbers are divisible by 7?

Q.23. Between 0 to 100, how many numbers are there lives remainder 2 if divided by 5?

Q. 24. The 5th and 14th terms of an AP are 22 and 58 respectively. Find out the AP and also find its 19th term.

Q. 25. The seventh term of an A.P. is 34 and 15^th term is 74. Determine the first term and the 40th term.

Q. 26. The 3rd and 12^th terms of an AP are 25 and -11 respectively. Find the first term and the common difference of the AP. Also, find its general term.

Q. 27. The first term of an A.P. is 5 and its 100^th term is – 292. Find the 50th term of this A.P.

Q. 28. The first term of an A.P. is -2 and the 10^th term is 16. Determine its 15^th term.

Q. 29. Determine the 25^th term of an A.P. whose 9th term is -6 and common difference is 5 / 4.

Q. 30. Determine the 2^nd term and r^th term of an A.P. whose 6th term is 12 and 8th term is 22.

Q.31. The fourth term of an A.P. is equal to 3 times the first term and the seventh term exceeds twice the third by 1. Find the first term and the common difference.

Q. 32. The third term of an A.P. is p and the fourth term is q. Find the 10th term.

Q. 33. Find the 15th term from the end of the AP 2, 6, 10, 14, …, 94.

Q. 34. For what value of n, the nth terms of the sequences 3, 10, 17, …. And 63, 65, 67,… are equal.

Q. 35. An A. P. consists of 21 terms. The sum of the three terms in the middle is 129 and of the last three is 237. Find the A.P.

Q. 36. Two A.P. have the same common difference. The first term of one of these is 13, and that of the other is 8. What is the difference between their:

1. 2^nd term?
2. 4^th term
3. 10^th term?
4. 30^th term?

Q.37. If a, b, c are in A.P. Prove that the following are also in A.P.
b + c, c + a, a + b

Q. 38. If 7 times the 7^th term of an A.P. is equal to 11 times its 11th term show that the 18^th term of the A.P. is zero.

Q. 39. If m times the mth term of an A.P. is equal to n times its nth term, prove that the
(m + n)th term of the A.P. is zero.

Q. 40. If 10 times the 10^th term of an AP is equal to 15 times its 15th term, show that its 25^th term is zero.

Q. 41. If the pth, qth and rth terms of an AP be a, b, c respectively then
show that a(q – r) + b(r – p) + c(p – q) = 0.

Q. 42. If the mth term of an AP be 1 / n and its nth term be 1 / m then show that its (mn)^th term is 1.

Q. 43. In an A.P., the 24^th term is twice the 10th term. Prove that the 36^th term is twice the 16^th term.

rth term from end, ar = l - ( r – 1 )d, where l is the last term, r is the position, d is the common difference.

Q. 44. Find 10^th term from the end

1. 5, 13, 21, ….is 181?
2. 5, 2, -1 ….is -22?
3. 2, 7, 12, 17,… is 87?
4. 21, 42, 63, 84, ….is 420?

Section - B

Arithmetic Progression ( Sum of n terms, Sn )

If the sum of 3, 4, 5 …..terms are given, then the terms should be taken in the form of.
for 3 terms…….. a – d, a , a + d
for 5 terms……..a – 2d, a – d, a, a + d, a + 2
So for odd number of terms ,common difference should be ‘d’
for 4 terms…….. a – 3d, a – d, a + d, a + 3d
So for even number of terms ,common difference should be ‘2d’

Q. 1. Divide 24 in three parts such that they are in AP and their product is 440.

Q. 2. Find three numbers in AP whose sum is 18 and whose product is 192.

Q. 3. The sum of the first tree terms of an A.P. is 36 while their product is 1620. Find the A.P.

Q. 4. In an A.P. the sum of three numbers is – 3 and their product is 8. Find the numbers

Q. 5. Find four terms in an A.P. whose sum is 20 and the sum of whose squares is 120.

Q. 6. The sum of three consecutive numbers in an A.P. is 27 and their product is 585. Find the numbers.

Q. 7. The four angles of a quadrilateral are in an A.P. whose common difference is 20. Find the angles.

Q. 8. Find five numbers in A.P. whose sum is 12 ½ and the ratio of first to the last is 2 : 3.

Sun of first ‘n’ terms in an A.P. Sn = n/2{ 2a + ( n – 1 )d }, where ‘n’ is the number of terms, ‘a’ is first term, ‘d’ is the common difference

Q. 9. Find the sum of the

1. first 11 terms of the AP: 2,6,10,…..
2. first 13 terms of the AP: -6, 0, 6, ………
3. first 20 terms of the AP: -1,-3,-5,-7,………
4. of first hundred even natural numbers divisible by 5.
5. of all odd numbers between 100 and 200.
6. of all even numbers between 5 and 100.
7. of all two-digit whole numbers divisible by 3.
8. of the series 1 + 5 + 9 + 3 + 8 + 13 + 5 + 11 + 16 + ……. upto 3n terms.
9. of all the three digit numbers which leave the remainder 2 when divided by 5.
10. of 2n terms of the series 12 – 22 + 32 – 42 + 52 – 62 + …..

If the last term is given then sum of first n terms, Sn = n/2 ( a + l ), where ‘n’ is the number of terms,‘a’ is the first term, and l is the last term.

Q. 10. Find the sum:

1. 70 + 68 + 66 + ….. + 40.
2. 2 + 4 + 6 + 8 +……+200
3. 3 + 11 + 19 + …….+ 803
4. –5 + (-8) + (-11) + ….+ (-230)

Q. 11. find the sum of n terms of the sequence {an} where an = 5 – 6n, n N.

Q. 12. find the sum of 32 terms of an A.P. whose third term is 1 and the 6th term is -11.

Q. 13. The sum of the 5^th and 7^th terms of an A.P. is 52 and the 10^th term is 46. Find the A.P.

Q. 14. The sum of the first 8 terms of an AP is 100 and the sum of its first 19 terms is 551. Find the first term and the common difference of the AP.

Q. 15. The sum of the first 30 terms of an AP is 1635. If its last term is 98, find the first term and the common difference of the given AP.

Q. 16. How many terms are there in an arithmetic sequence whose first and the fifth terms are – 14 and 2 respectively and the sum of terms is 40.

Q. 17. The last term of an A.P. is 252. Its first term and common difference are 12 and 6 respectively. Find the sum of the A.P. by (i) using last term (ii) without using last term. Verify that both the answers are same.

Q. 18. If the first term of an AP is 22, the common difference is -4 and the sum of the first n terms is 64, find n. Explain the double answer.

Q. 19. How many terms of the A.P. -6, -11 / 2, -5, …. Are needed to give the sum -25? Explain the double answer.

Q. 20. The sum of n terms of a sequence is 3n² + 4n. Find the nth term and show that the sequence is an A.P.

Q. 21. Find A.P., whose sum to n terms is 2n² + n.

Q. 22. How many terms of the AP 21, 18, 15, … must be added to get the sum 0?

Q. 23. The third term of an A.P. is 7, and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of the first 20 terms.

Q. 24. If the sum of the first p terms of an AP is the same as the sum of its first q terms (where p  q) then show that the sum of its first (p + q) terms is zero.

Q. 25. Show that the sum of first n even natural numbers is equal to (1 + 1/n) times the sum of the first n odd natural numbers.

Q. 26. If the pth term of an A.P. of 1 / q and the q^th term is 1 / p, show that the sum of pq terms is 1 / 2 (pq + 1)

Q. 27. A manufacturer of radio sets, produced 600 units in the third year and 700 units in the 7^th year. Assuming the production uniformly increases by a fixed number every year, find

1. the production in the first year(ii) the total production in 7 year
2. the production in the 10 th year