Mathematics · Textbook solutions

Sequences and Series

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 81 questions

8.2-8.3 Sequences and Series

17 q

Solved Examples

Worked · 3
  1. 8.1 Eg.1
    Write the first three terms in each of the following sequences defined by the following: (i) an=2n+5a_n = 2n + 5, (ii) an=n−34a_n = \frac{n-3}{4}.
  2. 8.1 Eg.2
    What is the 20th20^{th} term of the sequence defined by an=(n−1)(2−n)(3+n)a_n = (n-1)(2-n)(3+n) ?
  3. 8.1 Eg.3
    Let the sequence ana_n be defined as follows: a1=1a_1 = 1, an=an−1+2a_n = a_{n-1} + 2 for n≥2n \ge 2. Find first five terms and write corresponding series.

Exercise 8.1

Practice · 14
  1. Ex 8.1 Q1
    Write the first five terms of the sequence whose nthn^{th} term is an=n(n+2)a_n = n(n+2).
  2. Ex 8.1 Q2
    Write the first five terms of the sequence whose nthn^{th} term is an=nn+1a_n = \frac{n}{n+1}.
  3. Ex 8.1 Q3
    Write the first five terms of the sequence whose nthn^{th} term is an=2na_n = 2^n.
  4. Ex 8.1 Q4
    Write the first five terms of the sequence whose nthn^{th} term is an=2n−36a_n = \frac{2n-3}{6}.
  5. Ex 8.1 Q5
    Write the first five terms of the sequence whose nthn^{th} term is an=(−1)n−15n+1a_n = (-1)^{n-1} 5^{n+1}.
  6. Ex 8.1 Q6
    Write the first five terms of the sequence whose nthn^{th} term is an=nn2+54a_n = n\frac{n^2+5}{4}.
  7. Ex 8.1 Q7
    Find the indicated terms in the sequence whose nthn^{th} term is an=4n−3a_n = 4n - 3; a17a_{17}, a24a_{24}.
  8. Ex 8.1 Q8
    Find the indicated term in the sequence whose nthn^{th} term is an=n22na_n = \frac{n^2}{2^n}; a7a_7.
  9. Ex 8.1 Q9
    Find the indicated term in the sequence whose nthn^{th} term is an=(−1)n−1n3a_n = (-1)^{n-1} n^3; a9a_9.
  10. Ex 8.1 Q10
    Find the indicated term in the sequence whose nthn^{th} term is an=n(n−2)n+3a_n = \frac{n(n-2)}{n+3}; a20a_{20}.
  11. Ex 8.1 Q11
    Write the first five terms of the following sequence and obtain the corresponding series: a1=3a_1 = 3, an=3an−1+2a_n = 3a_{n-1} + 2 for all n>1n > 1.
  12. Ex 8.1 Q12
    Write the first five terms of the following sequence and obtain the corresponding series: a1=−1a_1 = -1, an=an−1na_n = \frac{a_{n-1}}{n}, n≥2n \ge 2.
  13. Ex 8.1 Q13
    Write the first five terms of the following sequence and obtain the corresponding series: a1=a2=2a_1 = a_2 = 2, an=an−1−1a_n = a_{n-1} - 1, n>2n > 2.
  14. Ex 8.1 Q14
    The Fibonacci sequence is defined by 1=a1=a21 = a_1 = a_2 and an=an−1+an−2a_n = a_{n-1} + a_{n-2}, n>2n > 2. Find an+1an\frac{a_{n+1}}{a_n}, for n=1,2,3,4,5n = 1, 2, 3, 4, 5

8.4 Geometric Progression

44 q

Solved Examples

Worked · 10
  1. 8.2 Eg.4
    Find the 10th10^{th} and nthn^{th} terms of the G.P. 5, 25, 125, ... .
  2. 8.2 Eg.5
    Which term of the G.P., 2, 8, 32, ... up to nn terms is 131072?
  3. 8.2 Eg.6
    In a G.P., the 3rd3^{rd} term is 24 and the 6th6^{th} term is 192. Find the 10th10^{th} term.
  4. 8.2 Eg.7
    Find the sum of first nn terms and the sum of first 5 terms of the geometric series 1+23+49+…1 + \frac{2}{3} + \frac{4}{9} + \ldots
  5. 8.2 Eg.8
    How many terms of the G.P. 3,32,34,…3, \frac{3}{2}, \frac{3}{4}, \ldots are needed to give the sum 3069512\frac{3069}{512} ?
  6. 8.2 Eg.9
    The sum of first three terms of a G.P. is 1312\frac{13}{12} and their product is −1-1. Find the common ratio and the terms.
  7. 8.2 Eg.10
    Find the sum of the sequence 7, 77, 777, 7777, ... to nn terms.
  8. 8.2 Eg.11
    A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
  9. 8.2 Eg.12
    Insert three numbers between 1 and 256 so that the resulting sequence is a G.P.
  10. 8.2 Eg.13
    If A.M. and G.M. of two positive numbers aa and bb are 10 and 8, respectively, find the numbers.

Exercise 8.2

Practice · 34
  1. Ex 8.2 Q1
    Find the 20th20^{th} and nthn^{th} terms of the G.P. 52,54,58,…\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots
  2. Ex 8.2 Q2
    Find the 12th12^{th} term of a G.P. whose 8th8^{th} term is 192 and the common ratio is 2.
  3. Ex 8.2 Q3
    The 5th5^{th}, 8th8^{th} and 11th11^{th} terms of a G.P. are pp, qq and ss, respectively. Show that q2=psq^2 = ps.
  4. Ex 8.2 Q4
    The 4th4^{th} term of a G.P. is square of its second term, and the first term is −3-3. Determine its 7th7^{th} term.
  5. Which term of the following sequences:
    Ex 8.2 Q5(a)
    2,22,4,…2, 2\sqrt{2}, 4, \ldots is 128 ?
  6. Ex 8.2 Q5(b)
    3,3,33,…\sqrt{3}, 3, 3\sqrt{3}, \ldots is 729 ?
  7. Ex 8.2 Q5(c)
    13,19,127,…\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots is 119683\frac{1}{19683} ?
  8. Ex 8.2 Q6
    For what values of xx, the numbers −27,x,−72-\frac{2}{7}, x, -\frac{7}{2} are in G.P.?
  9. Ex 8.2 Q7
    Find the sum to indicated number of terms in the geometric progression 0.15, 0.015, 0.0015, ... 20 terms.
  10. Ex 8.2 Q8
    Find the sum to indicated number of terms in the geometric progression 7\sqrt{7}, 21\sqrt{21}, 373\sqrt{7}, ... nn terms.
  11. Ex 8.2 Q9
    Find the sum to indicated number of terms in the geometric progression 1,−a,a2,−a3,…n1, -a, a^2, -a^3, \ldots n terms (if a≠−1a \ne -1).
  12. Ex 8.2 Q10
    Find the sum to indicated number of terms in the geometric progression x3,x5,x7,…nx^3, x^5, x^7, \ldots n terms (if x≠±1x \ne \pm 1).
  13. Ex 8.2 Q11
    Evaluate ∑k=111(2+3k)\sum_{k=1}^{11}(2 + 3^k).
  14. Ex 8.2 Q12
    The sum of first three terms of a G.P. is 3910\frac{39}{10} and their product is 1. Find the common ratio and the terms.
  15. Ex 8.2 Q13
    How many terms of G.P. 3,32,33,…3, 3^2, 3^3, \ldots are needed to give the sum 120?
  16. Ex 8.2 Q14
    The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to nn terms of the G.P.
  17. Ex 8.2 Q15
    Given a G.P. with a=729a = 729 and 7th7^{th} term 64, determine S7S_7.
  18. Ex 8.2 Q16
    Find a G.P. for which sum of the first two terms is −4-4 and the fifth term is 4 times the third term.
  19. Ex 8.2 Q17
    If the 4th4^{th}, 10th10^{th} and 16th16^{th} terms of a G.P. are xx, yy and zz, respectively. Prove that xx, yy, zz are in G.P.
  20. Ex 8.2 Q18
    Find the sum to nn terms of the sequence, 8, 88, 888, 8888... .
  21. Ex 8.2 Q19
    Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, 12\frac{1}{2}.
  22. Ex 8.2 Q20
    Show that the products of the corresponding terms of the sequences aa, arar, ar2ar^2, …arn−1\ldots ar^{n-1} and AA, ARAR, AR2AR^2, …ARn−1\ldots AR^{n-1} form a G.P, and find the common ratio.
  23. Ex 8.2 Q21
    Find four numbers forming a geometric progression in which the third term is greater than the first term by 9, and the second term is greater than the 4th4^{th} by 18.
  24. Ex 8.2 Q22
    If the pthp^{th}, qthq^{th} and rthr^{th} terms of a G.P. are aa, bb and cc, respectively. Prove that aq−r br−pcP−q=1a^{q-r}\, b^{r-p} c^{P-q} = 1.
  25. Ex 8.2 Q23
    If the first and the nthn^{th} term of a G.P. are aa and bb, respectively, and if P is the product of nn terms, prove that P2=(ab)nP^2 = (ab)^n.
  26. Ex 8.2 Q24
    Show that the ratio of the sum of first nn terms of a G.P. to the sum of terms from (n+1)th(n+1)^{th} to (2n)th(2n)^{th} term is 1rn\frac{1}{r^n}.
  27. Ex 8.2 Q25
    If aa, bb, cc and dd are in G.P. show that (a2+b2+c2)(b2+c2+d2)=(ab+bc+cd)2(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2.
  28. Ex 8.2 Q26
    Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
  29. Ex 8.2 Q27
    Find the value of nn so that an+1+bn+1an+bn\frac{a^{n+1} + b^{n+1}}{a^n + b^n} may be the geometric mean between aa and bb.
  30. Ex 8.2 Q28
    The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3+22):(3−22)\left(3 + 2\sqrt{2}\right) : \left(3 - 2\sqrt{2}\right).
  31. Ex 8.2 Q29
    If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A±(A+G)(A−G)A \pm \sqrt{(A+G)(A-G)}.
  32. Ex 8.2 Q30
    The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd2^{nd} hour, 4th4^{th} hour and nthn^{th} hour ?
  33. Ex 8.2 Q31
    What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?
  34. Ex 8.2 Q32
    If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.

Miscellaneous Exercise On Chapter 8

20 q

Solved Examples

Worked · 1
  1. Misc Eg.14
    If aa, bb, cc, dd and pp are different real numbers such that (a2+b2+c2)p2−2(ab+bc+cd)p+(b2+c2+d2)≤0(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \le 0, then show that aa, bb, cc and dd are in G.P.

Miscellaneous Exercise

Practice · 19
  1. Misc Q1
    If ff is a function satisfying f(x+y)=f(x)f(y)f(x+y) = f(x)f(y) for all x,y∈Nx, y \in \mathbf{N} such that f(1)=3f(1) = 3 and ∑x=1nf(x)=120\sum_{x=1}^{n} f(x) = 120, find the value of nn.
  2. Misc Q2
    The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
  3. Misc Q3
    The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
  4. Misc Q4
    The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.
  5. Misc Q5
    A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.
  6. Misc Q6
    If a+bxa−bx=b+cxb−cx=c+dxc−dx (x≠0)\frac{a+bx}{a-bx} = \frac{b+cx}{b-cx} = \frac{c+dx}{c-dx}\ (x \ne 0), then show that aa, bb, cc and dd are in G.P.
  7. Misc Q7
    Let S be the sum, P the product and R the sum of reciprocals of nn terms in a G.P. Prove that P2Rn=SnP^2R^n = S^n.
  8. Misc Q8
    If aa, bb, cc, dd are in G.P, prove that (an+bn)(a^n + b^n), (bn+cn)(b^n + c^n), (cn+dn)(c^n + d^n) are in G.P.
  9. Misc Q9
    If aa and bb are the roots of x2−3x+p=0x^2 - 3x + p = 0 and cc, dd are roots of x2−12x+q=0x^2 - 12x + q = 0, where aa, bb, cc, dd form a G.P. Prove that (q+p):(q−p)=17:15(q+p) : (q-p) = 17:15.
  10. Misc Q10
    The ratio of the A.M. and G.M. of two positive numbers aa and bb, is m:nm : n. Show that a:b=(m+m2−n2):(m−m2−n2)a : b = \left(m + \sqrt{m^2 - n^2}\right) : \left(m - \sqrt{m^2 - n^2}\right).
  11. Find the sum of the following series up to nn terms:
    Misc Q11(i)
    5+55+555+…5 + 55 + 555 + \ldots
  12. Misc Q11(ii)
    .6+.66+.666+….6 + .66 + .666 + \ldots
  13. Misc Q12
    Find the 20th20^{th} term of the series 2×4+4×6+6×8+…+n2 \times 4 + 4 \times 6 + 6 \times 8 + \ldots + n terms.
  14. Misc Q13
    A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much will the tractor cost him?
  15. Misc Q14
    Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
  16. Misc Q15
    A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th8^{th} set of letter is mailed.
  17. Misc Q16
    A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th15^{th} year since he deposited the amount and also calculate the total amount after 20 years.
  18. Misc Q17
    A manufacturer reckons that the value of a machine, which costs him Rs. 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
  19. Misc Q18
    150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.