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Mathematics · Textbook solutions

Sequences and Series

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 187 questions

2.3.1 General Term of a G.P.

11 q

Worked Example

Worked · 2
  1. Find nthn^{\text{th}} term of the following G.P.
    2.3.1a SolvedEx.1(i)
    3, −6, 12, −24, …3,\ -6,\ 12,\ -24,\ \ldots
  2. 2.3.1a SolvedEx.1(ii)
    5, 1, 15, 125, …5,\ 1,\ \frac{1}{5},\ \frac{1}{25},\ \ldots

Solved Examples

Worked · 9
  1. 2.3.1 SolvedEx.1
    Verify whether 1, −32, 94, −274, …1,\ \frac{-3}{2},\ \frac{9}{4},\ \frac{-27}{4},\ \ldots is a G.P., if it is a G.P. Find its ninth term.
  2. 2.3.1 SolvedEx.2
    Which term of the sequence 3, 3, 33, …\sqrt{3},\ 3,\ 3\sqrt{3},\ \ldots is 243?
  3. 2.3.1 SolvedEx.3
    For a G.P. If a=3a = 3 and t7=192t_7 = 192 find rr and t11t_{11}.
  4. 2.3.1 SolvedEx.4
    In a G.P., if the third term is 15\frac{1}{5} and sixth term is 1625\frac{1}{625}, find its nthn^{\text{th}} term.
  5. 2.3.1 SolvedEx.5
    If for a sequence {tn}\{t_n\}, tn=5n−24n−3t_n = \frac{5^{n-2}}{4^{n-3}} show that the sequence is a G.P. Find its first term and the common ratio.
  6. 2.3.1 SolvedEx.6
    Find three numbers in G.P. such that their sum is 42 and their product is 1728.
  7. 2.3.1 SolvedEx.7
    Find four numbers in G. P. such that their product is 64 and sum of the second and third number is 6.
  8. 2.3.1 SolvedEx.8
    If p, q, r, s are in G.P. then show that (q−r)2+(r−p)2+(s−q)2=(p−s)2(q-r)^2 + (r-p)^2 + (s-q)^2 = (p-s)^2
  9. 2.3.1 SolvedEx.9
    Shraddha deposited Rs. 8000 in a bank which pays annual interest rate of 8%. She kept it with the bank for 10 years with compound interest. Find the total amount she will receive after 10 years. [given (1.08)10=2.1575(1.08)^{10} = 2.1575]

Exercise 2.1

28 q
  1. Check whether the following sequences are G.P. If so, write tnt_n.
    Ex 2.1 Q1(i)
    2, 6, 18, 54, …2,\ 6,\ 18,\ 54,\ \ldots
  2. Ex 2.1 Q1(ii)
    1, −5, 25, −125 …1,\ -5,\ 25,\ -125\ \ldots
  3. Ex 2.1 Q1(iii)
    5, 15, 155, 1255, …\sqrt{5},\ \frac{1}{\sqrt{5}},\ \frac{1}{5\sqrt{5}},\ \frac{1}{25\sqrt{5}},\ \ldots
  4. Ex 2.1 Q1(iv)
    3, 4, 5, 6, …3,\ 4,\ 5,\ 6,\ \ldots
  5. Ex 2.1 Q1(v)
    7, 14, 21, 28, …7,\ 14,\ 21,\ 28,\ \ldots
  6. For the G.P.
    Ex 2.1 Q2(i)
    If r=13r = \frac{1}{3}, a=9a = 9 find t7t_7
  7. Ex 2.1 Q2(ii)
    If a=7243a = \frac{7}{243}, r=3r = 3 find t6t_6.
  8. Ex 2.1 Q2(iii)
    If r=−3r = -3 and t6=1701t_6 = 1701, find aa.
  9. Ex 2.1 Q2(iv)
    If a=23a = \frac{2}{3}, t6=162t_6 = 162, find rr.
  10. Ex 2.1 Q3
    Which term of the G.P. 5, 25, 125, 625, …5,\ 25,\ 125,\ 625,\ \ldots is 5105^{10}?
  11. Ex 2.1 Q4
    For what values of xx, the terms 43, x, 427\frac{4}{3},\ x,\ \frac{4}{27} are in G.P?
  12. Ex 2.1 Q5
    If for a sequence, tn=5n−32n−3t_n = \frac{5^{n-3}}{2^{n-3}}, show that the sequence is a G.P. Find its first term and the common ratio.
  13. Ex 2.1 Q6
    Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
  14. Ex 2.1 Q7
    Find four numbers in G.P. such that sum of the middle two numbers is 103\frac{10}{3} and their product is 1.
  15. Ex 2.1 Q8
    Find five numbers in G. P. such that their product is 1024 and fifth term is square of the third term.
  16. Ex 2.1 Q9
    The fifth term of a G.P. is xx, eighth term of a G.P. is yy and eleventh term of a G.P. is zz verify whether y2=xzy^2 = xz.
  17. Ex 2.1 Q10
    If p,q,r,sp, q, r, s are in G.P. show that p+qp+q, q+rq+r, r+sr+s are also in G.P.
  18. Ex 2.1 Q11
    The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th5^{\text{th}} hour ?
  19. Ex 2.1 Q12
    A ball is dropped from a height of 80 ft. The ball is such that it rebounds (34)th\left(\frac{3}{4}\right)^{\text{th}} of the height it has fallen. How high does the ball rebound on 6th6^{\text{th}} bounce? How high does the ball rebound on nthn^{\text{th}} bounce?
  20. The numbers 33, xx and x+6x + 6 are in G.P. Find
    Ex 2.1 Q13(i)
    xx
  21. Ex 2.1 Q13(ii)
    20th20^{\text{th}} term
  22. Ex 2.1 Q13(iii)
    nthn^{\text{th}} term.
  23. Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after
    Ex 2.1 Q14(i)
    3 years
  24. Ex 2.1 Q14(ii)
    10 years
  25. Ex 2.1 Q14(iii)
    nn years.
  26. The numbers x−6x - 6, 2x2x and x2x^2 are in G.P. Find
    Ex 2.1 Q15(i)
    xx
  27. Ex 2.1 Q15(ii)
    1st1^{\text{st}} term
  28. Ex 2.1 Q15(iii)
    nthn^{\text{th}} term.

2.3.2 Sum of the First n Terms of a G.P.

14 q

Solved Examples

Worked · 14
  1. 2.3.2 SolvedEx.1
    If a=1a = 1, r=2r = 2 find SnS_n for the G.P.
  2. 2.3.2 SolvedEx.2
    For a G.P. 0.02, 0.04, 0.08, 0.016, …0.02,\ 0.04,\ 0.08,\ 0.016,\ \ldots, find SnS_n.
  3. 2.3.2 SolvedEx.3
    For the following G.P. 3, −3, 3, −3, …3,\ -3,\ 3,\ -3,\ \ldots, find SnS_n.
  4. 2.3.2 SolvedEx.4
    For a G.P. if a=6a = 6, r=2r = 2, find S10S_{10}.
  5. 2.3.2 SolvedEx.5
    How many terms of G.P. 2, 22, 23, 24, …2,\ 2^2,\ 2^3,\ 2^4,\ \ldots are needed to give the sum 2046.
  6. 2.3.2 SolvedEx.6
    If for a G.P. r=2r = 2, S10=1023S_{10} = 1023, find aa.
  7. 2.3.2 SolvedEx.7
    For a G.P. a=3a = 3, r=2r = 2, Sn=765S_n = 765, find nn.
  8. 2.3.2 SolvedEx.8
    For a G.P. if S3=16S_3 = 16, S6=144S_6 = 144, find the first term and the common ratio of the G.P.
  9. 2.3.2 SolvedEx.9
    Find 5+55+555+5555+…5 + 55 + 555 + 5555 + \ldots upto nn terms.
  10. 2.3.2 SolvedEx.10
    Find the sum to nn terms 0.3+0.33+0.333+…0.3 + 0.33 + 0.333 + \ldots nn terms
  11. 2.3.2 SolvedEx.11
    Find the nthn^{\text{th}} term of the sequence 0.4, 0.44, 0.444,…0.4,\ 0.44,\ 0.444, \ldots
  12. 2.3.2 SolvedEx.12
    For a sequence, if Sn=5(4n−1)S_n = 5(4^n - 1), find the nthn^{\text{th}} term, hence verify that it is a G.P., Also find rr.
  13. 2.3.2 SolvedEx.13
    A teacher wanted to reward a student by giving some chocolates. He gave the student two choices. He could either have 60 chocolates at once or he could get 1 chocolate on the first day, 2 on the second day, 4 on the third day and so on for 6 days. Which option should the student choose to get more chocolates?
  14. 2.3.2 SolvedEx.14
    Mr. Pritesh got a job with an annual salary package of Rs. 4,00,000 with 10% annual increment. Find his salary in the 5th5^{\text{th}} year and also find his total earnings through salary in 10 years. [Given (1.1)4=1.4641(1.1)^4 = 1.4641, (1.1)10=2.59374(1.1)^{10} = 2.59374]

Exercise 2.2

23 q
  1. For the following G.P.s, find SnS_n
    Ex 2.2 Q1(i)
    3, 6, 12, 24, …3,\ 6,\ 12,\ 24,\ \ldots
  2. Ex 2.2 Q1(ii)
    p, q, q2p, q3p2, …p,\ q,\ \frac{q^2}{p},\ \frac{q^3}{p^2},\ \ldots
  3. Ex 2.2 Q1(iii)
    0.7, 0.07, 0.007, …0.7,\ 0.07,\ 0.007,\ \ldots
  4. Ex 2.2 Q1(iv)
    5, −5, 55, −25…\sqrt{5},\ -5,\ 5\sqrt{5},\ -25 \ldots
  5. For a G.P.
    Ex 2.2 Q2(i)
    a=2a = 2, r=−23r = -\frac{2}{3}, find S6S_6
  6. Ex 2.2 Q2(ii)
    If S5=1023S_5 = 1023, r=4r = 4, Find aa
  7. For a G.P.
    Ex 2.2 Q3(i)
    If a=2a = 2, r=3r = 3, Sn=242S_n = 242 find nn.
  8. Ex 2.2 Q3(ii)
    For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of rr.
  9. For a G.P.
    Ex 2.2 Q4(i)
    If t3=20t_3 = 20, t6=160t_6 = 160, find S7S_7
  10. Ex 2.2 Q4(ii)
    If t4=16t_4 = 16, t9=512t_9 = 512, find S10S_{10}
  11. Find the sum to nn terms
    Ex 2.2 Q5(i)
    3+33+333+3333+…3 + 33 + 333 + 3333 + \ldots
  12. Ex 2.2 Q5(ii)
    8+88+888+8888+…8 + 88 + 888 + 8888 + \ldots
  13. Find the sum to nn terms
    Ex 2.2 Q6(i)
    0.4+0.44+0.444+…0.4 + 0.44 + 0.444 + \ldots
  14. Ex 2.2 Q6(ii)
    0.7+0.77+0.777+…0.7 + 0.77 + 0.777 + \ldots
  15. Find the sum to nn terms of the sequence
    Ex 2.2 Q7(i)
    0.5, 0.05, 0.005, …0.5,\ 0.05,\ 0.005,\ \ldots
  16. Ex 2.2 Q7(ii)
    0.2, 0.02, 0.002, …0.2,\ 0.02,\ 0.002,\ \ldots
  17. Ex 2.2 Q8
    For a sequence, if Sn=2(3n−1)S_n = 2(3^n - 1), find the nthn^{\text{th}} term, hence show that the sequence is a G.P.
  18. Ex 2.2 Q9
    If S,P,RS, P, R are the sum, product and sum of the reciprocals of nn terms of a G.P. respectively, then verify that [SR]n=P2\left[\frac{S}{R}\right]^n = P^2.
  19. Ex 2.2 Q10
    If Sn, S2n, S3nS_n,\ S_{2n},\ S_{3n} are the sum of n,2n,3nn, 2n, 3n terms of a G.P. respectively, then verify that Sn(S3n−S2n)=(S2n−Sn)2S_n(S_{3n} - S_{2n}) = (S_{2n} - S_n)^2.
  20. Find
    Ex 2.2 Q11(i)
    ∑r=110(3×2r)\sum_{r=1}^{10} (3 \times 2^r)
  21. Ex 2.2 Q11(ii)
    ∑r=1105×3r\sum_{r=1}^{10} 5 \times 3^r
  22. Ex 2.2 Q12
    The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is Rs. 15 Lac. [Given : (1.05)5=1.28(1.05)^5 = 1.28, (1.05)6=1.34(1.05)^6 = 1.34]
  23. Ex 2.2 Q13
    If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5=1.47][(1.08)^5 = 1.47]

2.4 Sum of Infinite Terms of a G.P.

4 q

Worked Example

Worked · 1
  1. 2.4a SolvedEx.1
    1+12+14+18+116+…1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots

Solved Examples

Worked · 3
  1. Determine whether the sum of all the terms in the series is finite. In case it is finite find it.
    2.4 SolvedEx.1(i)
    13,132,133,…\frac{1}{3}, \frac{1}{3^{2}}, \frac{1}{3^{3}}, \ldots
  2. 2.4 SolvedEx.1(ii)
    35,−925,27125,−81625,…\frac{3}{5}, \frac{-9}{25}, \frac{27}{125}, \frac{-81}{625}, \ldots
  3. 2.4 SolvedEx.1(iii)
    1,−3,9,−27,81,…1, -3, 9, -27, 81, \ldots

2.4.1 Recurring Decimals as Rational Numbers

3 q

Solved Examples

Worked · 3
  1. Express the following recurring decimals as rational numbers, using the sum to infinite terms of a G.P.
    2.4.1 SolvedEx.1(i)
    0.66666…0.66666\ldots
  2. 2.4.1 SolvedEx.1(ii)
    0.46‾0.\overline{46}
  3. 2.4.1 SolvedEx.1(iii)
    2.5‾2.\overline{5}

Exercise 2.3

19 q
  1. Determine whether the sum to infinity of the following G.P.s exist, if exists find them
    Ex 2.3 Q1(i)
    12,14,18,116,…\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots
  2. Ex 2.3 Q1(ii)
    2,43,89,1627,…2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, \ldots
  3. Ex 2.3 Q1(iii)
    −3,1,−13,19,…-3, 1, \frac{-1}{3}, \frac{1}{9}, \ldots
  4. Ex 2.3 Q1(iv)
    15,−25,45,−85,165,…\frac{1}{5}, \frac{-2}{5}, \frac{4}{5}, \frac{-8}{5}, \frac{16}{5}, \ldots
  5. Ex 2.3 Q1(v)
    9,8.1,7.29,…9, 8.1, 7.29, \ldots
  6. Express the following recurring decimals as a rational number.
    Ex 2.3 Q2(i)
    0.7‾0.\overline{7}
  7. Ex 2.3 Q2(ii)
    2.4‾2.\overline{4}
  8. Ex 2.3 Q2(iii)
    2.35‾2.3\overline{5}
  9. Ex 2.3 Q2(iv)
    51.02‾51.0\overline{2}
  10. Ex 2.3 Q3
    If the common ratio of a G.P. is 23\frac{2}{3} and sum to infinity is 12. Find the first term.
  11. Ex 2.3 Q4
    If the first term of the G.P. is 16 and its sum to infinity is 9617\frac{96}{17} find the common ratio.
  12. Ex 2.3 Q5
    The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
  13. Find
    Ex 2.3 Q6(i)
    ∑r=1∞4(0.5)r\sum_{r=1}^{\infty} 4(0.5)^{r}
  14. Ex 2.3 Q6(ii)
    ∑r=1∞(−13)r\sum_{r=1}^{\infty}\left(-\frac{1}{3}\right)^{r}
  15. Ex 2.3 Q6(iii)
    ∑r=0∞(−8)(−12)r\sum_{r=0}^{\infty}(-8)\left(-\frac{1}{2}\right)^{r}
  16. Ex 2.3 Q6(iv)
    ∑n=1∞0.4n\sum_{n=1}^{\infty} 0.4^{n}
  17. The mid points of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of
    Ex 2.3 Q7(i)
    the areas of all the squares
  18. Ex 2.3 Q7(ii)
    the perimeters of all the square
  19. Ex 2.3 Q8
    A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball.

2.5 Harmonic Progression

2 q

Solved Examples

Worked · 2
  1. 2.5 SolvedEx.1
    Find the nthn^{\text{th}} term of the H.P. 12,25,13,27…\frac{1}{2}, \frac{2}{5}, \frac{1}{3}, \frac{2}{7} \ldots
  2. 2.5 SolvedEx.2
    Find the nthn^{\text{th}} term of H.P. 15,1,−13,−17,…\frac{1}{5}, 1, \frac{-1}{3}, \frac{-1}{7}, \ldots

2.6.3 Harmonic Mean

5 q

Solved Examples

Worked · 5
  1. 2.6.3 SolvedEx.1
    Find A.M., G.M., H.M. of the numbers 4 and 16
  2. 2.6.3 SolvedEx.2
    Insert 4 terms between 2 and 22 so that the new sequence is in AP.
  3. 2.6.3 SolvedEx.3
    Insert two numbers between 29\frac{2}{9} and 112\frac{1}{12} so that the resulting sequence is a HP.
  4. 2.6.3 SolvedEx.4
    Insert two numbers between 1 and 27 so that the resulting sequence is a G.P.
  5. 2.6.3 SolvedEx.5
    The A.M. of two numbers exceeds their G.M. by 2 and their H.M. by 185\frac{18}{5}. Find the Numbers.

Exercise 2.4

13 q
  1. Verify whether the following sequences are H.P.
    Ex 2.4 Q1(i)
    13,15,17,19,…\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \ldots
  2. Ex 2.4 Q1(ii)
    13,16,112,124,…\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, \ldots
  3. Ex 2.4 Q1(iii)
    5,1017,1032,1047,…5, \frac{10}{17}, \frac{10}{32}, \frac{10}{47}, \ldots
  4. Find the nthn^{\text{th}} term and hence find the 8th8^{\text{th}} term of the following HPs
    Ex 2.4 Q2(i)
    12,15,18,111,…\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \frac{1}{11}, \ldots
  5. Ex 2.4 Q2(ii)
    14,16,18,110,…\frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \frac{1}{10}, \ldots
  6. Ex 2.4 Q2(iii)
    15,110,115,120,…\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}, \ldots
  7. Ex 2.4 Q3
    Find A.M. of two positive numbers whose G.M. and H. M. are 4 and 165\frac{16}{5} respectively.
  8. Ex 2.4 Q4
    Find H.M. of two positive numbers A.M. and G.M. are 152\frac{15}{2} and 6
  9. Ex 2.4 Q5
    Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
  10. Ex 2.4 Q6
    Insert two numbers between 14\frac{1}{4} and 13\frac{1}{3} so that the resulting sequence is a HP.
  11. Ex 2.4 Q7
    Insert two numbers between 1 and −27-27 so that the resulting sequence is a G.P.
  12. Ex 2.4 Q8
    If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by 185\frac{18}{5}, find the numbers.
  13. Ex 2.4 Q9
    Find two numbers whose A.M. exceeds their GM by 12\frac{1}{2} and their HM by 2526\frac{25}{26}

2.7.1 Sum of n Terms of an A.G.P.

1 q

Solved Example

Worked · 1
  1. 2.7.1 SolvedEx.1
    Find tnt_n and the sum of nn terms of 1,4,12,32,80,192,…1, 4, 12, 32, 80, 192, \ldots

Exercise 2.5

7 q
  1. Find SnS_n of the following arithmetico – geometric sequences
    Ex 2.5 Q1(i)
    2,4x,6x2,8x3,10x4,…2, 4x, 6x^2, 8x^3, 10x^4, \ldots
  2. Ex 2.5 Q1(ii)
    1,4x,7x2,10x3,13x4,…1, 4x, 7x^2, 10x^3, 13x^4, \ldots
  3. Ex 2.5 Q1(iii)
    1,2×3,3×9,4×27,5×81,…1, 2 \times 3, 3 \times 9, 4 \times 27, 5 \times 81, \ldots
  4. Ex 2.5 Q1(iv)
    3,12,36,96,240,…3, 12, 36, 96, 240, \ldots
  5. Find the sum to infinity of the following arithmetico – geometric sequence
    Ex 2.5 Q2(i)
    1,24,316,464,…1, \frac{2}{4}, \frac{3}{16}, \frac{4}{64}, \ldots
  6. Ex 2.5 Q2(ii)
    3,65,925,12125,15625,…3, \frac{6}{5}, \frac{9}{25}, \frac{12}{125}, \frac{15}{625}, \ldots
  7. Ex 2.5 Q2(iii)
    1,−43,79,−1027,…1, \frac{-4}{3}, \frac{7}{9}, \frac{-10}{27}, \ldots

Properties of Summation

4 q

Solved Examples

Worked · 4
  1. 2.7.2 SolvedEx.1
    Evaluate ∑r=1n(8r−7)\displaystyle\sum_{r=1}^{n} (8r - 7)
  2. 2.7.2 SolvedEx.2
    Find 32+42+52+…+2923^{2} + 4^{2} + 5^{2} + \ldots + 29^{2}.
  3. 2.7.2 SolvedEx.3
    Find 1002−992+982−972+…+22−12100^{2} - 99^{2} + 98^{2} - 97^{2} + \ldots + 2^{2} - 1^{2}
  4. 2.7.2 SolvedEx.4
    Find 1×5+3×7+5×9+7×11…1 \times 5 + 3 \times 7 + 5 \times 9 + 7 \times 11 \ldots upto nn terms.

Exercise 2.6

10 q
  1. Ex 2.6 Q1
    Find the sum ∑r=1n(r+1)(2r−1)\displaystyle\sum_{r=1}^{n} (r+1)(2r-1)
  2. Ex 2.6 Q2
    Find ∑r=1n(3r2−2r+1)\displaystyle\sum_{r=1}^{n} (3r^{2} - 2r + 1)
  3. Ex 2.6 Q3
    Find ∑r=1n(1+2+3+…+rr)\displaystyle\sum_{r=1}^{n} \left(\dfrac{1 + 2 + 3 + \ldots + r}{r}\right)
  4. Ex 2.6 Q4
    Find ∑r=1n(13+23+…+r3r(r+1))\displaystyle\sum_{r=1}^{n} \left(\dfrac{1^{3} + 2^{3} + \ldots + r^{3}}{r(r+1)}\right)
  5. Ex 2.6 Q5
    Find the sum 5×7+9×11+13×15+…5 \times 7 + 9 \times 11 + 13 \times 15 + \ldots upto nn terms.
  6. Ex 2.6 Q6
    Find the sum 22+42+62+82+…2^{2} + 4^{2} + 6^{2} + 8^{2} + \ldots upto nn terms.
  7. Ex 2.6 Q7
    Find (702−692)+(682−672)+(662−652)+…+(22−12)(70^{2} - 69^{2}) + (68^{2} - 67^{2}) + (66^{2} - 65^{2}) + \ldots + (2^{2} - 1^{2})
  8. Ex 2.6 Q8
    Find the sum 1×3×5+3×5×7+5×7×9+…+(2n−1)(2n+1)(2n+3)1 \times 3 \times 5 + 3 \times 5 \times 7 + 5 \times 7 \times 9 + \ldots + (2n-1)(2n+1)(2n+3)
  9. Ex 2.6 Q9
    If 1×2+2×3+3×4+4×5+… upto n terms1+2+3+4+… upto n terms=1003\dfrac{1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + \ldots \text{ upto } n \text{ terms}}{1 + 2 + 3 + 4 + \ldots \text{ upto } n \text{ terms}} = \dfrac{100}{3}, find nn.
  10. Ex 2.6 Q10
    If S1S_{1}, S2S_{2} and S3S_{3} are the sums of first nn natural numbers, their squares and their cubes respectively then show that 9S22=S3(1+8S1)9 S_{2}^{2} = S_{3}(1 + 8 S_{1}).

Miscellaneous Exercise 2

43 q

(I) Select the correct answer

Practice · 10
  1. Misc I Q1
    The common ratio for the G.P. 0.12,0.24,0.48,0.12, 0.24, 0.48, is –
    1. A.
      0.120.12
    2. B.
      0.20.2
    3. C.
      0.020.02
    4. D.
      22
  2. Misc I Q2
    The tenth term of the geometric sequence 14,−12,1,−2,…\frac{1}{4}, \frac{-1}{2}, 1, -2, \ldots is –
    1. A.
      10241024
    2. B.
      11024\frac{1}{1024}
    3. C.
      −128-128
    4. D.
      −1128\frac{-1}{128}
  3. Misc I Q3
    If for a G.P. t6t3=145854\frac{t_6}{t_3} = \frac{1458}{54} then r=?r = ?
    1. A.
      33
    2. B.
      22
    3. C.
      11
    4. D.
      −1-1
  4. Misc I Q4
    Which term of the geometric progression 1,2,4,8,…1, 2, 4, 8, \ldots is 2048.
    1. A.
      10th10^{\text{th}}
    2. B.
      11th11^{\text{th}}
    3. C.
      12th12^{\text{th}}
    4. D.
      13th13^{\text{th}}
  5. Misc I Q5
    If common ratio of the G.P is 5, 5th5^{\text{th}} term is 1875, the first term is -
    1. A.
      33
    2. B.
      55
    3. C.
      1515
    4. D.
      −5-5
  6. Misc I Q6
    The sum of 3 terms of a G.P. is 214\frac{21}{4} and their product is 1 then the common ratio is –
    1. A.
      11
    2. B.
      22
    3. C.
      44
    4. D.
      88
  7. Misc I Q7
    Sum to infinity of a G.P. 5,−52,54,−58,516,…5, -\frac{5}{2}, \frac{5}{4}, -\frac{5}{8}, \frac{5}{16}, \ldots is –
    1. A.
      55
    2. B.
      −12-\frac{1}{2}
    3. C.
      103\frac{10}{3}
    4. D.
      310\frac{3}{10}
  8. Misc I Q8
    The tenth term of H.P. 29,17,219,112,…\frac{2}{9}, \frac{1}{7}, \frac{2}{19}, \frac{1}{12}, \ldots is -
    1. A.
      127\frac{1}{27}
    2. B.
      92\frac{9}{2}
    3. C.
      52\frac{5}{2}
    4. D.
      2727
  9. Misc I Q9
    Which of the following is not true, where A, G, H are the AM, GM, HM of aa and bb respectively. (a,b>0)(a, b > 0)
    1. A.
      A=a+b2A = \frac{a+b}{2}
    2. B.
      G=abG = \sqrt{ab}
    3. C.
      H=2aba+bH = \frac{2ab}{a+b}
    4. D.
      A=GHA = GH
  10. Misc I Q10
    The G.M. of two numbers exceeds their H.M. by 65\frac{6}{5}, the A.M. exceeds G.M. by 32\frac{3}{2} the two numbers are ...
    1. A.
      6,1526, \frac{15}{2}
    2. B.
      15,2515, 25
    3. C.
      3,123, 12
    4. D.
      65,32\frac{6}{5}, \frac{3}{2}

(II) Answer the following

Practice · 33
  1. Misc II Q1
    In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term.
  2. Misc II Q2
    Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is 23\frac{2}{3}
  3. Misc II Q3
    For a G.P. a=43a = \frac{4}{3} and t7=2431024t_7 = \frac{243}{1024}, find the value of rr.
  4. Misc II Q4
    For a sequence, if tn=5n−27n−3t_n = \frac{5^{n-2}}{7^{n-3}}, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
  5. Misc II Q5
    Find three numbers in G.P. such that their sum is 35 and their product is 1000.
  6. Misc II Q6
    Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.
  7. Misc II Q7
    For a sequence Sn=4(7n−1)S_n = 4(7^n - 1) verify that the sequence is a G.P.
  8. Misc II Q8
    Find 2+22+222+2222+…2 + 22 + 222 + 2222 + \ldots upto nn terms.
  9. Misc II Q9
    Find the nthn^{\text{th}} term of the sequence 0.6,0.66,0.666,0.6666,…0.6, 0.66, 0.666, 0.6666, \ldots
  10. Misc II Q10
    Find ∑r=1n(5r2+4r−3)\sum_{r=1}^{n} (5r^2 + 4r - 3)
  11. Misc II Q11
    Find ∑r=1nr(r−3)(r−2)\sum_{r=1}^{n} r(r-3)(r-2)
  12. Misc II Q12
    Find ∑r=1n12+22+32+…+r22r+1\sum_{r=1}^{n} \frac{1^2 + 2^2 + 3^2 + \ldots + r^2}{2r+1}
  13. Misc II Q13
    Find ∑r=1n13+23+33+…r3(r+1)2\sum_{r=1}^{n} \frac{1^3 + 2^3 + 3^3 + \ldots r^3}{(r+1)^2}
  14. Misc II Q14
    Find 2×6+4×9+6×12+…2 \times 6 + 4 \times 9 + 6 \times 12 + \ldots upto nn terms.
  15. Misc II Q15
    Find 2×5×8+4×7×10+6×9×12+…2 \times 5 \times 8 + 4 \times 7 \times 10 + 6 \times 9 \times 12 + \ldots upto nn terms.
  16. Misc II Q16
    Find 121+12+222+12+22+323+…\frac{1^2}{1} + \frac{1^2 + 2^2}{2} + \frac{1^2 + 2^2 + 3^2}{3} + \ldots upto nn terms.
  17. Misc II Q17
    Find 122+132+142+152+…20212^2 + 13^2 + 14^2 + 15^2 + \ldots 20^2
  18. Misc II Q18
    If 1+2+3+4+5+… upto n terms1×2+2×3+3×4+4×5+… upto n terms=322\frac{1 + 2 + 3 + 4 + 5 + \ldots \text{ upto n terms}}{1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + \ldots \text{ upto n terms}} = \frac{3}{22} Find the value of nn.
  19. Misc II Q19
    Find (502−492)+(482−472)+(462−452)+…+(22−12)(50^2 - 49^2) + (48^2 - 47^2) + (46^2 - 45^2) + \ldots + (2^2 - 1^2).
  20. Misc II Q20
    If 1×3+2×5+3×7+… upto n terms13+23+33+… upto n terms=59\frac{1 \times 3 + 2 \times 5 + 3 \times 7 + \ldots \text{ upto n terms}}{1^3 + 2^3 + 3^3 + \ldots \text{ upto n terms}} = \frac{5}{9}, find the value of nn.
  21. Misc II Q21
    For a G.P. if t2=7t_2 = 7, t4=1575t_4 = 1575 find aa
  22. Misc II Q22
    If for a G.P. t3=13t_3 = \frac{1}{3}, t6=181t_6 = \frac{1}{81} find rr
  23. Misc II Q23
    Find ∑r=1n(23)r\sum_{r=1}^{n} \left(\frac{2}{3}\right)^r.
  24. Misc II Q24
    Find kk so that k−1,k,k+2k-1, k, k+2 are consecutive terms of a G.P.
  25. Misc II Q25
    If for a G.P. first term is (27)2(27)^2 and seventh term is (8)2(8)^2, find S8S_8.
  26. Misc II Q26
    If pthp^{\text{th}}, qthq^{\text{th}} and rthr^{\text{th}} terms of a G.P. are x,y,zx, y, z respectively. Find the value of xq−r⋅yr−p⋅zp−qx^{q-r} \cdot y^{r-p} \cdot z^{p-q}
  27. Misc II Q27
    Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
  28. Misc II Q28
    If p,q,rp, q, r are in G.P. and p1/x=q1/y=r1/zp^{1/x} = q^{1/y} = r^{1/z}, verify whether x,y,zx, y, z are in A.P. or G.P. or neither.
  29. Misc II Q29
    If a,b,ca, b, c are in G.P. and ax2+2bx+c=0ax^2 + 2bx + c = 0 and px2+2qx+r=0px^2 + 2qx + r = 0 have common roots then verify that pb2−2qba+ra2=0p b^2 - 2 q b a + r a^2 = 0
  30. Misc II Q30
    If p,q,r,sp, q, r, s are in G.P., show that (p2+q2+r2)(q2+r2+s2)=(pq+qr+rs)2(p^2 + q^2 + r^2)(q^2 + r^2 + s^2) = (pq + qr + rs)^2
  31. Misc II Q31
    If p,q,r,sp, q, r, s are in G.P., show that (pn+qn)(p^n + q^n), (qn+rn)(q^n + r^n), (rn+sn)(r^n + s^n) are also in G.P.
  32. Misc II Q32
    Find the coefficient of x6x^6 in the expansion of e2xe^{2x} using series expansion.
  33. Misc II Q33
    Find the sum of infinite terms of 1+45+725+10125+13625+…1 + \frac{4}{5} + \frac{7}{25} + \frac{10}{125} + \frac{13}{625} + \ldots