TMUA Sequences & Series 2 ā Questions and Answers
Question 1: Which formula gives the sum of the first n natural numbers?
- n(n+1)/2 (Correct answer)
- n(nā1)/2
- nĀ²/2
- n(n+2)/2
Correct answer: n(n+1)/2
The sum 1 + 2 + 3 + ... + n equals n(n+1)/2, a standard result.
Question 2: The sum to infinity of a geometric series is 12 and the first term is 4. What is the common ratio?
- 1/2
- 1/3
- 2/3 (Correct answer)
- 3/4
Correct answer: 2/3
Using Sā = a/(1ār): 12 = 4/(1ār) gives 1ār = 1/3, so r = 2/3.
Question 3: Find the 15th term of the arithmetic sequence: 7, 10, 13, 16, ...
- 46
- 49 (Correct answer)
- 52
- 43
Correct answer: 49
T_15 = 7 + (14)(3) = 7 + 42 = 49.
Question 4: Which of the following infinite geometric series converges?
- 2 + 4 + 8 + ...
- 3 ā 6 + 12 ā ...
- 5 + 5/2 + 5/4 + ... (Correct answer)
- 1 + 1 + 1 + ...
Correct answer: 5 + 5/2 + 5/4 + ...
The series 5 + 5/2 + 5/4 + ... has ratio r = 1/2, and |r| < 1 guarantees convergence.
Question 5: If the sum ā_{k=1}^{n} k = 210, what is the value of n?
- 18
- 19
- 20 (Correct answer)
- 21
Correct answer: 20
Using n(n+1)/2 = 210 gives n(n+1) = 420; since 20 Ć 21 = 420, n = 20.
Question 6: A sequence is defined by aā = 2 and a_{n+1} = 3a_n ā 1. What is the value of aā?
- 38
- 41 (Correct answer)
- 44
- 35
Correct answer: 41
aā = 5, aā = 14, aā = 3(14) ā 1 = 41.
Question 7: The sum of the first 5 terms of a geometric sequence with common ratio 2 is 31. What is the first term?
- 1 (Correct answer)
- 2
- 3
- 4
Correct answer: 1
Sā = a(2āµ ā 1)/(2 ā 1) = 31a = 31, so a = 1.
TMUA Sequences & Series 2
Which formula gives the sum of the first n natural numbers?