Sequences and Series Class 11 Notes

• A finite sequence has a fixed number of terms; an infinite sequence continues indefinitely.
• A series converges if its partial sums approach a finite limit; otherwise it diverges.
• The two most common progressions are arithmetic (AP) and geometric (GP).

• Three numbers in AP can be written as (a − d), a, (a + d) to simplify calculations.
• Arithmetic mean of a and b is
• $A=\frac{a+b}{2}$
• , which is the single term inserted between them to form an AP.

• Three numbers in GP can be written as a/r, a, ar.
• Geometric mean of a and b is
• $G=\sqrt{ab}$
• , which is the single term inserted between them to form a GP.
• The sum to infinity exists only when |r| < 1 — the series converges.

• If AM and GM of two numbers are given, the numbers are the roots of
• $x^2-2(\text{AM})x+(\text{GM}{)}^2=0$
• .
• For three numbers: AM = (a + b + c)/3 and GM =
• $\sqrt[3]{abc}$
• .

• The sum of cubes equals the square of the sum of the first n natural numbers.
• These formulas are used to convert special series into closed-form expressions for quick evaluation.
• They are derived by induction or by algebraic manipulation of known identities.

• Adding or subtracting a constant from every term of an AP preserves the common difference but shifts the sum.
• Multiplying or dividing every term of an AP by a non-zero constant creates a new AP with the same number of terms but a scaled common difference.
• In a GP, multiplying or dividing every term by a non-zero constant preserves the common ratio.
• The product of corresponding terms of two APs does not form an AP, but the product of corresponding terms of two GPs with the same ratio is again a GP.
• If a, b, c are in AP then 2b = a + c; if a, b, c are in GP then b² = ac.

Example: Find the 20th term and the sum of 20 terms of the AP: 3, 7, 11, 15, ...

Solution: a = 3, d = 4. $t_{20}=3+19\times 4=79$. $S_{20}=\frac{20}{2}(3+79)=10\times 82=820$.

Example: The sum of two numbers is 14 and their product is 45. Find the numbers using AM and GM.

Solution: AM = 14/2 = 7, GM = √45 = 3√5. The numbers are the roots of $x^2-14x+45=0$, which factors as (x − 5)(x − 9) = 0. The numbers are 5 and 9.

Exercises (NCERT Q1–Q46)

• 1) Write the first five terms of the sequence whose nth term is aₙ = n(n + 2).
• 2) Write the first five terms of the sequence whose nth term is aₙ = n/(n + 1).
• 3) Write the first five terms of the sequence whose nth term is aₙ = 2ⁿ.
• 4) Write the first five terms of the sequence whose nth term is aₙ = (2n − 3)/6.
• 5) Write the first five terms of the sequence whose nth term is aₙ = (−1)ⁿ⁻¹·5ⁿ⁺¹.
• 6) Write the first five terms of the sequence whose nth term is aₙ = n(n² + 5)/4.
• 7) Find the 17th and 24th terms of the sequence whose nth term is aₙ = 4n − 3.
• 8) Find the 7th term of the sequence whose nth term is aₙ = n²/2ⁿ.
• 9) Find the 9th term of the sequence whose nth term is aₙ = (−1)ⁿ⁻¹·n³.
• 10) Find the 20th term of the sequence whose nth term is aₙ = n(n − 2)/(n + 3).
• 11) Write the first five terms of the sequence and obtain the corresponding series: a₁ = 3, aₙ = 3aₙ₋₁ + 2 for all n > 1.
• 12) Write the first five terms of the sequence and obtain the corresponding series: a₁ = −1, aₙ = aₙ₋₁/n, n ≥ 2.
• 13) Write the first five terms of the sequence and obtain the corresponding series: a₁ = a₂ = 2, aₙ = aₙ₋₁ − 1, n > 2.
• 14) The Fibonacci sequence is defined by 1 = a₁ = a₂ and aₙ = aₙ₋₁ + aₙ₋₂, n > 2. Find aₙ₊₁/aₙ for n = 1, 2, 3, 4, 5.
• 15) Find the 20th and nth terms of the G.P. 5/2, 5/4, 5/8, ….
• 16) Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.
• 17) The 5th, 8th and 11th terms of a G.P. are p, q and s respectively. Show that q² = ps.
• 18) The 4th term of a G.P. is the square of its second term, and the first term is −3. Determine its 7th term.
• 19) Which term of the following sequences (i) 2, 2√2, 4, … is 128? (ii) √3, 3, 3√3, … is 729? (iii) 1/3, 1/9, 1/27, … is 1/19683?
• 20) For what values of x are the numbers −2/7, x, −7/2 in G.P.?
• 21) Find the sum to 20 terms of the sequence 0.15, 0.015, 0.0015, ….
• 22) Find the sum to n terms of the sequence √7, √21, 3√7, ….
• 23) Find the sum to n terms of the sequence 1, −a, a², −a³, … (if a ≠ −1).
• 24) Find the sum to n terms of the sequence x³, x⁵, x⁷, … (if x ≠ ±1).
• 25) Evaluate Σ(k = 1 to 11) (2 + 3ᵏ).
• 26) The sum of the first three terms of a G.P. is 39/10 and their product is 1. Find the common ratio and the terms.
• 27) How many terms of the G.P. 3, 3², 3³, … are needed to give the sum 120?
• 28) The sum of the 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.
• 29) Given a G.P. with a = 729 and 7th term 64, determine S₇.
• 30) 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.
• 31) 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.
• 32) Find the sum to n terms of the sequence 8, 88, 888, 8888, ….
• 33) 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.
• 34) Show that the products of the corresponding terms of the sequences a, ar, ar², …, arⁿ⁻¹ and A, AR, AR², …, ARⁿ⁻¹ form a G.P. and find the common ratio.
• 35) Find four numbers forming a G.P. in which the third term is greater than the first by 9, and the second term is greater than the 4th by 18.
• 36) If the pth, qth and rth terms of a G.P. are a, b and c respectively, prove that a^(q−r) · b^(r−p) · c^(p−q) = 1.
• 37) If the first and nth terms of a G.P. are a and b respectively, and P is the product of n terms, prove that P² = (ab)ⁿ.
• 38) Show that the ratio of the sum of the first n terms of a G.P. to the sum of terms from the (n+1)th to the (2n)th term is 1/rⁿ.
• 39) If a, b, c and d are in G.P., show that (a² + b² + c²)(b² + c² + d²) = (ab + bc + cd)².
• 40) Insert two numbers between 3 and 81 so that the resulting sequence is a G.P.
• 41) Find the value of n so that (aⁿ⁺¹ + bⁿ⁺¹)/(aⁿ + bⁿ) may be the geometric mean between a and b.
• 42) 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).
• 43) If A and G are the A.M. and G.M. respectively between two positive numbers, prove that the numbers are A ± √((A + G)(A − G)).
• 44) The number of bacteria in a culture doubles every hour. If 30 bacteria are present originally, how many will be present at the end of the 2nd hour, 4th hour and nth hour?
• 45) What will Rs 500 amount to in 10 years after its deposit in a bank which pays an annual interest rate of 10% compounded annually?
• 46) If the A.M. and G.M. of the roots of a quadratic equation are 8 and 5 respectively, obtain the quadratic equation.

Answers at a glance

1. 1
   Q1 → 3, 8, 15, 24, 35.
2. 2
   Q2 → 1/2, 2/3, 3/4, 4/5, 5/6.
3. 3
   Q3 → 2, 4, 8, 16, 32.
4. 4
   Q4 → −1/6, 1/6, 1/2, 5/6, 7/6.
5. 5
   Q5 → 25, −125, 625, −3125, 15625.
6. 6
   Q6 → 3/2, 9/2, 21/2, 21, 75/2.
7. 7
   Q7 → 65, 93.
8. 8
   Q8 → 49/128.
9. 9
   Q9 → 729.
10. 10
    Q10 → 360/23.
11. 11
    Q11 → 3, 11, 35, 107, 323; series 3 + 11 + 35 + 107 + 323 + ….
12. 12
    Q12 → −1, −1/2, −1/6, −1/24, −1/120; series −1 − 1/2 − 1/6 − 1/24 − 1/120.
13. 13
    Q13 → 2, 2, 1, 0, −1; series 2 + 2 + 1 + 0 − 1.
14. 14
    Q14 → 1, 2, 3/2, 5/3, 8/5.
15. 15
    Q15 → 5/2²⁰ and 5/2ⁿ.
16. 16
    Q16 → 3072.
17. 17
    Q17 → q² = ps (shown).
18. 18
    Q18 → −2187.
19. 19
    Q19 → (i) 13th, (ii) 12th, (iii) 9th.
20. 20
    Q20 → x = ±1.
21. 21
    Q21 → (1/6)(1 − 10⁻²⁰).
22. 22
    Q22 → √7(3ⁿᐟ² − 1)/(√3 − 1).
23. 23
    Q23 → (1 − (−a)ⁿ)/(1 + a).
24. 24
    Q24 → x³(x²ⁿ − 1)/(x² − 1).
25. 25
    Q25 → 265741.
26. 26
    Q26 → r = 5/2 or 2/5; terms (2/5, 1, 5/2) or (5/2, 1, 2/5).
27. 27
    Q27 → 4 terms.
28. 28
    Q28 → a = 16/7, r = 2, Sₙ = (16/7)(2ⁿ − 1).
29. 29
    Q29 → 2059.
30. 30
    Q30 → a = −4/3 with r = 2, or a = 4 with r = −2.
31. 31
    Q31 → Proved (y² = xz).
32. 32
    Q32 → 80(10ⁿ − 1)/81 − 8n/9.
33. 33
    Q33 → 496.
34. 34
    Q34 → G.P. with common ratio rR (shown).
35. 35
    Q35 → 3, −6, 12, −24.
36. 36
    Q36 → Proved: the product equals 1.
37. 37
    Q37 → P² = (ab)ⁿ (shown).
38. 38
    Q38 → Ratio = 1/rⁿ (shown).
39. 39
    Q39 → Identity proved.
40. 40
    Q40 → 9 and 27.
41. 41
    Q41 → n = −1/2.
42. 42
    Q42 → Ratio is (3 + 2√2) : (3 − 2√2).
43. 43
    Q43 → Proved: numbers are A ± √((A+G)(A−G)).
44. 44
    Q44 → 120, 480, and 30·2ⁿ.
45. 45
    Q45 → 500(1.1)¹⁰ ≈ Rs 1296.87.
46. 46
    Q46 → x² − 16x + 25 = 0.

MCQ Questions

1 Mark Questions

1. 1
   Find the 10th term of the AP 2, 5, 8, … (Ans: 2 + 9×3 = 29)
2. 2
   Insert three A.M.s between 2 and 18. (Ans: 6, 10, 14)
3. 3
   Find the 5th term of the GP 1, 2, 4, 8, … (Ans: 16)
4. 4
   Write the sum formula for the first n terms of a GP. (Ans: Sn = a(rⁿ − 1)/(r − 1) for r ≠ 1)

2 Mark Questions

1. 1
   Find the sum 1 + 2 + 3 + … + 100. (Ans: 100×101/2 = 5050)
2. 2
   The A.M. of two numbers is 10 and their difference is 8; find the numbers. (Ans: 14 and 6)
3. 3
   Find the sum of the first 5 terms of the GP 1, 2, 4, 8, … (Ans: 31)

3 Mark Questions

1. 1
   If the 4th term of an AP is 12 and the 10th term is 30, find the first term and common difference. (Ans: a = 0, d = 3)
2. 2
   Find the sum to n terms of the series 3, 6, 12, 24, … (Ans: 3(2ⁿ − 1))
3. 3
   If a, b, c are in GP, prove that b² = ac, and verify b = 6 for a = 4, c = 9.

4 Mark Questions

1. 1
   Prove that the sum of the first n natural numbers is n(n + 1)/2, and use it to find the sum of all natural numbers from 51 to 100. (Ans: 5050 − 1275 = 3775)
2. 2
   The sum of three numbers in AP is 15 and their product is 80; find the numbers. (Ans: 2, 5, 8 — taking a − d, a, a + d)