Chapter-5 (Sequence and Series) Notes

Sequence and Series: Notes

Mathematics • Grade 10 • Sequence and Series

1. Sequence and Series

A list of numbers in a definite order following a rule is a sequence. Adding its terms gives a series.

	Arithmetic (AP)	Geometric (GP)
Rule	Same difference d = t₂ − t₁	Same ratio r = t₂ ÷ t₁
General term	tₙ = a + (n − 1)d	tₙ = arⁿ⁻¹
Example	2, 4, 6, 8, …	2, 6, 18, 54, …

Review. Classify and find the 6th and 8th terms.

(a) 2, 4, 6, 8: AP, d = 2, tₙ = 2n; t₆ = 12, t₈ = 16.
(b) 1, 3, 5, 7: AP, d = 2, tₙ = 2n − 1; t₆ = 11, t₈ = 15.
(c) 1, 4, 16, 64: GP, r = 4, tₙ = 4ⁿ⁻¹; t₆ = 1024, t₈ = 16384.
(d) 24, 12, 6, 3: GP, r = ½, tₙ = 24(½)ⁿ⁻¹; t₆ = 3/4, t₈ = 3/16.
(e) 5, 11, 8, 14: neither (differences +6, −3 alternate).
(f) 2, 6, 18, 54: GP, r = 3, tₙ = 2·3ⁿ⁻¹; t₆ = 486, t₈ = 4374.
Series, e.g. (a): 2 + 4 + 6 + 8 + …

2. Arithmetic Means

• Terms between the first and last terms of an AP are arithmetic means.
• One mean between a and b: m = (a + b)/2
• Many means: total terms n = (number of means) + 2. Use b = a + (n − 1)d to find d, then add d repeatedly.

Example 1. AM between 7 and 17.

m = (7 + 17)/2 = 12

Example 2. Three AMs between 3 and 23.

n = 5: 23 = 3 + 4d ⇒ d = 5. Means: 8, 13, 18

Example 3. Seven AMs between 3 and 43.

n = 9: 43 = 3 + 8d ⇒ d = 5. Means: 8, 13, 18, 23, 28, 33, 38 (the book writes t₉ = 23 by mistake)

3. Sum of Arithmetic Series

• Sₙ = n/2 [2a + (n − 1)d]
• When the last term l is known: Sₙ = n/2 (a + l)
• Sum of first n natural numbers = n(n + 1)/2; first n odd numbers = n²; first n even numbers = n(n + 1).

Derivation: Write the sum forwards and backwards and add: 2Sₙ = n(a + l).

Activity. Incentives 800, 1000, 1200, … (d = 200). Total for 13 months?

t₁₃ = 800 + 12 × 200 = Rs. 3,200. S₁₃ = 13/2 [1600 + 2400] = Rs. 26,000

Example 4. Sum of first 20 terms of 1 + 2 + 3 + …

S₂₀ = 10 [2 + 19] = 210

Example 5. Sum of first 13 terms of 2 + 4 + 6 + …

S₁₃ = 13/2 [4 + 24] = 182

Example 6. Sum of first 28 terms of 5 + 17 + 29 + …

d = 12. S₂₈ = 14 [10 + 324] = 4676

Example 7. a = 2, last term 29, Sₙ = 155. Find n and d.

155 = n/2 × 31 ⇒ n = 10. 29 = 2 + 9d ⇒ d = 3

Example 8. How many terms of 24 + 20 + 16 + … sum to 72?

72 = n/2 [48 − 4(n − 1)] ⇒ n² − 13n + 36 = 0 ⇒ (n − 9)(n − 4) = 0 ⇒ n = 4 or 9 (terms after the 5th cancel out)

Example 9. t₃ = 0, t₁₀ = 42. Find S₁₅.

7d = 42 ⇒ d = 6; a + 12 = 0 ⇒ a = −12. S₁₅ = 15/2 [−24 + 84] = 450

Example 10. Salary Rs. 40,000, Rs. 1,000 grade each year. Income in 6 years? Extra needed for Rs. 35,00,000?

a = 4,80,000, d = 12,000. S₆ = 3 [9,60,000 + 60,000] = Rs. 30,60,000. Extra = 35,00,000 − 30,60,000 = Rs. 4,40,000

4. Geometric Means

• Terms between the first and last terms of a GP are geometric means.
• One mean between a and b: m = √(ab)
• Many means: n = (means) + 2; use b = arⁿ⁻¹ to find r, then multiply repeatedly.

Example 1. GM between 2 and 32.

m = √64 = 8

Example 2. Three GMs between 5 and 405.

405 = 5r⁴ ⇒ r⁴ = 81 ⇒ r = 3. Means: 15, 45, 135

5. Sum of Geometric Series

• If r > 1: Sₙ = a(rⁿ − 1)/(r − 1)
• If r < 1: Sₙ = a(1 − rⁿ)/(1 − r)
• When the last term l is known: Sₙ = (lr − a)/(r − 1)

Example 3. Sum of first 5 terms of 1 + 3 + 9 + …

S₅ = (3⁵ − 1)/2 = 121

Example 4. Sum of first 5 terms of 2 + 1 + ½ + ¼ + …

r = ½: S₅ = 2(1 − 1/32)/(1/2) = 31/8

Example 5. 3 + 6 + 12 + … + 768.

r = 2: S = (768 × 2 − 3)/(2 − 1) = 1533

Example 6. a = 7, l = 448, S = 889. Find r.

889(r − 1) = 448r − 7 ⇒ 441r = 882 ⇒ r = 2

Example 7. t₃ = 27, t₆ = 729. Find S₁₀.

r³ = 27 ⇒ r = 3; 9a = 27 ⇒ a = 3. S₁₀ = 3(3¹⁰ − 1)/2 = 88,572

Example 8. How many terms of 64 + 32 + 16 + … sum to 255/2?

128(1 − (½)ⁿ) = 255/2 ⇒ (½)ⁿ = 1/256 ⇒ n = 8

Example 9. Rs. 19,682 repaid in 9 instalments, each 3 times the previous. Difference of first and last?

a(3⁹ − 1)/2 = 19682 ⇒ a = 2. t₉ = 2 × 3⁸ = 13122. Difference = Rs. 13,120

Exam tips: (1) Identify AP or GP first. (2) Write a, d or r, n clearly. (3) Use simultaneous equations when two terms are given. (4) In word problems, change monthly figures to yearly before applying the formula.

Sequence and Series • Grade 10 Mathematics