AIME Sequences and Series 2

Question 1

The 3rd and 6th terms of a geometric sequence are 8 and 64. Find the common ratio.

Accepted Answer

2

Answer

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.

Accepted Answer

25

Answer

101=5+(n-1)×4 gives n-1=24, so n=25.

Question 3

Find the sum 1+2+3+…+100.

Accepted Answer

5050

Answer

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.

Accepted Answer

3/4

Answer

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?

Accepted Answer

34

Answer

Continuing: 1,1,2,3,5,8,13,21,34; the 9th term is 34.

AIME - American Invitational Mathematics Examination Practice Test