1950 AMC 12 Problem 39

Question

Given the series

2+1+\dfrac12+\dfrac14+\cdots

and the following five statements:

(1)

the sum increases without limit.

(2)

the sum decreases without limit.

(3)

the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small.

(4)

the difference between the sum and

4

can be made less than any positive quantity no matter how small.

(5)

the sum approaches a limit.

Of these statements, the correct ones are:

Only

3

and

4

Only

5

Only

2

and

4

Only

2,

3,

and

4

Only

4

and

5

Answer: E

Concepts:geometric sequence summation calculus

Difficulty rating: 1670

Small hint:

Distinguish the terms of the sequence from its partial sums

Big hint:

A geometric series with first term

2

and ratio

\frac{1}{2}

has partial sums approaching

4

Solution:

The partial sums increase toward

\frac{2}{1-\frac12}=4,

so they neither increase nor decrease without limit. This makes statements

1

and

2

false, while statements

4

and

5

are true.

In statement

3,

“any term” refers to an already selected term, whose distance from zero is fixed and cannot be made smaller; one can instead choose a sufficiently late term below any prescribed positive bound. Thus statement

3

, as worded, is false.

Therefore only statements

4

and

5

are correct, so the correct answer is E.

1950 AMC 12 Problem 39

Answer: E

Difficulty rating: 1670

Small hint:

Big hint:

Solution:

Problem 39 in other years