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Solutions

Overview

Step-by-step NCERT solutions for Sequences and Series (Chapter 8, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Sequences and Series textbook chapter.

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All 64 questions in Sequences and Series are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 8.1

  1. Write the first five terms of the sequence whose nth term is: a_n = n(n + 2)
  2. Write the first five terms of the sequence whose nth term is: a_n = n/(n + 1)
  3. Write the first five terms of the sequence whose nth term is: a_n = 2^n
  4. Write the first five terms of the sequence whose nth term is: a_n = (2n – 3)/6
  5. Write the first five terms of the sequence whose nth term is: a_n = (–1)^(n–1) × 5^(n+1)
  6. Write the first five terms of the sequence whose nth term is: a_n = n(n² + 5)/4
  7. Find a_17 and a_24 for the sequence whose nth term is: a_n = 4n – 3
  8. Find a_7 for the sequence whose nth term is: a_n = n²/2^n
  9. Find a_9 for the sequence whose nth term is: a_n = (–1)^(n–1) n³
  10. Find a_20 for the sequence whose nth term is: a_n = n(n – 2)/(n + 3)
  11. Write the first five terms of the sequence defined by a_1 = 3, a_n = 3a_(n–1) + 2 for all n > 1, and obtain the corresponding series.
  12. Write the first five terms of the sequence defined by a_1 = –1, a_n = a_(n–1)/n for n ≥ 2, and obtain the corresponding series.
  13. Write the first five terms of the sequence defined by a_1 = a_2 = 2, a_n = a_(n–1) – 1 for n > 2, and obtain the corresponding series.
  14. The Fibonacci sequence is defined by 1 = a_1 = a_2 and a_n = a_(n–1) + a_(n–2) for n > 2. Find a_(n+1)/a_n for n = 1, 2, 3, 4, 5.

Exercise 8.2

  1. Find the 20th and nth terms of the G.P.: 5/2, 5/4, 5/8, ...
  2. Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
  3. The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q² = ps.
  4. The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7th term.
  5. Which term of the following sequences:
    • (a) 2, 2√2, 4, ... is 128?
    • (b) √3, 3, 3√3, ... is 729?
    • (c) 1/3, 1/9, 1/27, ... is 1/19683?
  6. For what values of x, the numbers –2/7, x, –7/2 are in G.P.?
  7. Find the sum to 20 terms of the G.P.: 0.15, 0.015, 0.0015, ...
  8. Find the sum to n terms of the G.P.: √7, √21, 3√7, ...
  9. Find the sum to n terms of the G.P.: 1, –a, a², –a³, ... (if a ≠ –1)
  10. Find the sum to n terms of the G.P.: x³, x⁵, x⁷, ... (if x ≠ ±1)
  11. Evaluate: Σ (2 + 3^k) for k = 1 to 11
  12. The sum of first three terms of a G.P. is 39/10 and their product is 1. Find the common ratio and the terms.
  13. How many terms of G.P. 3, 3², 3³, … are needed to give the sum 120?
  14. 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 n terms of the G.P.
  15. Given a G.P. with a = 729 and 7th term 64, determine S_7.
  16. Find a G.P. for which the sum of the first two terms is –4 and the fifth term is 4 times the third term.
  17. If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
  18. Find the sum to n terms of the sequence: 8, 88, 888, 8888, ...
  19. Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2.
  20. Show that the products of the corresponding terms of the sequences a, ar, ar², …ar^(n–1) and A, AR, AR², …AR^(n–1) form a G.P., and find the common ratio.
  21. 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 4th by 18.
  22. If the p^th, q^th and r^th terms of a G.P. are a, b and c, respectively. Prove that a^(q–r) b^(r–p) c^(p–q) = 1.
  23. If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P² = (ab)^n.
  24. Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n+1)^th to (2n)^th term is 1/r^n.
  25. If a, b, c and d are in G.P., show that (a² + b² + c²)(b² + c² + d²) = (ab + bc + cd)².
  26. Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
  27. Find the value of n so that (a^(n+1) + b^(n+1))/(a^n + b^n) may be the geometric mean between a and b.
  28. The sum of two numbers is 6 times their geometric mean. Show that the numbers are in the ratio (3 + 2√2) : (3 – 2√2).
  29. 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)).
  30. 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 2nd hour, 4th hour and nth hour?
  31. What will Rs 500 amount to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?
  32. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.

Miscellaneous Exercise

  1. If f is a function satisfying f(x + y) = f(x)f(y) for all x, y ∈ N such that f(1) = 3 and Σ f(x) = 120 (x from 1 to n), find the value of n.
  2. 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. 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 the G.P.
  4. 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. 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. If (a + bx)/(a – bx) = (b + cx)/(b – cx) = (c + dx)/(c – dx) (x ≠ 0), then show that a, b, c and d are in G.P.
  7. Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P²R^n = S^n.
  8. If a, b, c, d are in G.P., prove that (a^n + b^n), (b^n + c^n), (c^n + d^n) are in G.P.
  9. If a and b are the roots of x² – 3x + p = 0 and c, d are roots of x² – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17 : 15.
  10. The ratio of the A.M. and G.M. of two positive numbers a and b is m : n. Show that a : b = (m + √(m² – n²)) : (m – √(m² – n²)).
  11. Find the sum of the following series up to n terms:
    • (i) 5 + 55 + 555 + …
    • (ii) 0.6 + 0.66 + 0.666 + …
  12. Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + ... + n terms.
  13. 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?
  14. Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
  15. A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail it 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 the 8th set of letters is mailed.
  16. A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in the 15th year since he deposited the amount and also calculate the total amount after 20 years.
  17. 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.
  18. 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.
Sample

Worked examples

3 questions from Sequences and Series, solved step by step.

Exercise 8.1

Write the first five terms of the sequence whose nth term is: aₙ = n(n + 2)

Substitute n = 1, 2, 3, 4, 5 in aₙ = n(n + 2):

a₁ = 1(1 + 2) = 1 × 3 = 3

a₂ = 2(2 + 2) = 2 × 4 = 8

a₃ = 3(3 + 2) = 3 × 5 = 15

a₄ = 4(4 + 2) = 4 × 6 = 24

a₅ = 5(5 + 2) = 5 × 7 = 35

Answer: 3, 8, 15, 24, 35

Exercise 8.2

Find the 20th and nth terms of the G.P.: 5/2, 5/4, 5/8, ...

Here a = 5/2 and r = (5/4)/(5/2) = 1/2.

The nth term of a G.P. is aₙ = ar^(n–1).

nth term: aₙ = (5/2) × (1/2)^(n–1) = 5/2^n

20th term: a_20 = (5/2) × (1/2)¹⁹ = 5/(2 × 2¹⁹) = 5/2²⁰

Answer: aₙ = 5/2^n, a_20 = 5/2²⁰

Miscellaneous Exercise

If f is a function satisfying f(x + y) = f(x)f(y) for all x, y ∈ N such that f(1) = 3 and Σ f(x) = 120 (x from 1 to n), find the value of n.

Since f(x + y) = f(x)f(y) for all x, y ∈ N and f(1) = 3:

f(2) = f(1+1) = f(1)f(1) = 9 = 3²

f(3) = f(2+1) = f(2)f(1) = 27 = 3³

In general, f(n) = 3^n.

The sum Σ_{x=1}^n f(x) = 3 + 3² + 3³ + ... + 3^n

= 3(3^n – 1)/(3 – 1) = 3(3^n – 1)/2 = 120

⇒ 3^n – 1 = 80

⇒ 3^n = 81 = 3⁴

⇒ n = 4

Answer: n = 4

All 64 questions in Sequences and Series are solved step by step in the Solutions PDF above.

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