PRE CALCULUS Pre-Calculus Sequences and Series 5 — Questions and Answers
Question 1: An arithmetic sequence has a₁ = 8 and a₁₀ = 44. What is the common difference?
- 3
- 4 (Correct answer)
- 5
- 6
Correct answer: 4
d = (a₁₀ − a₁)/(10−1) = (44−8)/9 = 36/9 = 4.
Question 2: Which series diverges?
- Σ(1/2)ⁿ
- Σ(0.9)ⁿ
- Σ(1.1)ⁿ (Correct answer)
- Σ(0.01)ⁿ
Correct answer: Σ(1.1)ⁿ
A geometric series diverges when |r| ≥ 1; here r = 1.1 > 1 causes divergence.
Question 3: Find the sum of the arithmetic series: 3 + 7 + 11 + … + 47.
- 275
- 300 (Correct answer)
- 325
- 350
Correct answer: 300
n = (47−3)/4 + 1 = 12; S = 12(3+47)/2 = 12·25 = 300.
Question 4: What does the index of summation represent in sigma notation Σ(k=1 to n) aₖ?
- The number of terms being added
- The variable that takes integer values from 1 to n (Correct answer)
- The common ratio of the sequence
- The first term of the series
Correct answer: The variable that takes integer values from 1 to n
The index k is the variable that steps through integer values from 1 to n, selecting each term.
Question 5: Two consecutive terms of a geometric sequence are 48 and 12. What is the common ratio?
- −1/4
- 1/4 (Correct answer)
- 1/3
- 4
Correct answer: 1/4
r = 12/48 = 1/4.
Question 6: A geometric series has 6 terms with a first term of 1 and last term of 243. What is the common ratio?
- 2
- 3 (Correct answer)
- 4
- 5
Correct answer: 3
a₆ = 1·r⁵ = 243 → r⁵ = 243 → r = 3.
Question 7: The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, … What is the 10th term?
- 34
- 55 (Correct answer)
- 89
- 144
Correct answer: 55
Continuing: 13, 21, 34, 55 — the 10th term is 55.
PRE CALCULUS Pre-Calculus Sequences and Series 5
An arithmetic sequence has a₁ = 8 and a₁₀ = 44.
What is the common difference?