AIME Sequences and Series 2 — Questions and Answers
Question 1: The 3rd and 6th terms of a geometric sequence are 8 and 64. Find the common ratio.
- 2 (Correct answer)
- 4
- 3
- 8
Correct answer: 2
a_6/a_3=r^3=64/8=8 so r=2.
Question 2: Find the number of terms in the arithmetic sequence 5, 9, 13, …, 101.
- 25 (Correct answer)
- 20
- 30
- 24
Correct answer: 25
101=5+(n-1)×4 gives n-1=24, so n=25.
Question 3: Find the sum 1+2+3+…+100.
- 5050 (Correct answer)
- 5000
- 4950
- 5100
Correct answer: 5050
S=n(n+1)/2=100×101/2=5050.
Question 4: If the first term of a geometric series is 1/2 and the sum to infinity is 2, find the common ratio.
- 3/4 (Correct answer)
- 1/2
- 1/4
- 1/3
Correct answer: 3/4
S=a/(1-r)=2 so 1/2/(1-r)=2, giving 1-r=1/4, r=3/4.
Question 5: In the Fibonacci sequence 1,1,2,3,5,8,13,…, what is the 9th term?
- 34 (Correct answer)
- 21
- 55
- 13
Correct answer: 34
Continuing: 1,1,2,3,5,8,13,21,34; the 9th term is 34.
Question 6: For the sequence a_n = n^2 - n, find a_7 - a_5.
- 22 (Correct answer)
- 20
- 24
- 18
Correct answer: 22
a_7=49-7=42, a_5=25-5=20; difference=22.
AIME Sequences and Series 2
The 3rd and 6th terms of a geometric sequence are 8 and 64.
Find the common ratio.