Chapter-5 (Sequence and Series) Important Questions

Sequence and Series: Important Questions

Mathematics • Grade 10 • Sequence and Series

A. Theory

2 marks1. What are arithmetic means? Express m in terms of a and b if a, m, b are in AP.

Terms between the first and last terms of an AP. m = (a + b)/2

3 marks2. Derive the formula Sₙ = n/2 [2a + (n − 1)d].

Sₙ = a + (a + d) + … + l and Sₙ = l + (l − d) + … + a. Adding, 2Sₙ = n(a + l) ⇒ Sₙ = n/2 (a + l) = n/2 [a + a + (n − 1)d] = n/2 [2a + (n − 1)d]

2 marks3. What are geometric means? Express m if a, m, b are in GP.

Terms between the first and last terms of a GP. m = √(ab)

3 marks4. Write the formulas for the sum of a GP when r > 1, r < 1 and when the last term is given.

a(rⁿ − 1)/(r − 1); a(1 − rⁿ)/(1 − r); (lr − a)/(r − 1)

B. Numerical

3 marks5. Find 7 arithmetic means between 3 and 43.

d = 5: 8, 13, 18, 23, 28, 33, 38

3 marks6. Find 6 arithmetic means between 70 and 14.

d = −8: 62, 54, 46, 38, 30, 22

3 marks7. If x + 1, x + 5, 3x + 1 are in AP, find x.

4 = 2x − 4 ⇒ x = 4

4 marks8. Find the sum of 7 + 11 + 15 + … to 20 terms.

S₂₀ = 10 [14 + 76] = 900

4 marks9. Find the sum of natural numbers from 50 to 100.

n = 51: S = 51/2 (150) = 3825

4 marks10. How many terms of 4 + 10 + 16 + … have sum 374?

3n² + n − 374 = 0 ⇒ n = 11

4 marks11. How many terms of 24 + 20 + 16 + … have sum 72?

n² − 13n + 36 = 0 ⇒ n = 4 or 9

5 marks12. Find S₁₅ if t₃ = 0 and t₁₀ = 42.

d = 6, a = −12: S₁₅ = 450

5 marks13. A person earns Rs. 40,000 a month and gets a Rs. 1,000 monthly grade every year. Total income in 6 years? Extra needed to earn Rs. 35,00,000?

Rs. 30,60,000; Rs. 4,40,000

3 marks14. Find the GM between 2 and 32, and 3 GMs between 5 and 405.

8; 15, 45, 135

4 marks15. Find 4 GMs between 6 and 192.

r⁵ = 32, r = 2: 12, 24, 48, 96

4 marks16. Find the sum of 2 + 4 + 8 + … to 6 terms and of 1/9 + 1/3 + 1 + … to 5 terms.

126; 121/9

4 marks17. Find the sum of 3 + 6 + 12 + … + 768.

1533

5 marks18. If t₃ = 27 and t₆ = 729 of a GP, find S₁₀.

r = 3, a = 3: S₁₀ = 88,572

5 marks19. Sarita borrowed Rs. 43,680 and pays it back in 6 instalments, each three times the previous. Find the difference of first and last.

a = 120, last = 29,160: Rs. 29,040

Sequence and Series • Grade 10 Mathematics