AIME Sequences and Series 3

Question 1

Find the sum of the geometric series 3 + 3/2 + 3/4 + … to infinity.

Accepted Answer

6

Answer

S=3/(1-1/2)=3/(1/2)=6.

Question 2

An arithmetic sequence has S_n = 3n^2 + 2n. Find the 5th term.

Accepted Answer

27

Answer

a_5=S_5-S_4=(75+10)-(48+8)=85-56=29; recheck: S_5=3(25)+10=85, S_4=3(16)+8=56, a_5=29.

Question 3

The sum of an infinite geometric series is 12 and the first term is 3. Find the common ratio.

Accepted Answer

3/4

Answer

12=3/(1-r), so 1-r=3/12=1/4, r=3/4.

Question 4

How many terms of the series 1+3+5+7+… are needed to reach a sum of 81?

Accepted Answer

9

Answer

Sum of first n odd numbers=n^2; n^2=81 gives n=9.

Question 5

Find the value of the sum Σ(k=1 to 5) k(k+1).

Accepted Answer

70

Answer

k=1:2, k=2:6, k=3:12, k=4:20, k=5:30; sum=2+6+12+20+30=70.

AIME - American Invitational Mathematics Examination Practice Test