1955 AMC 12 Problem 45

Question

Given a geometric sequence with the first term

\ne0

and

r\ne0

and an arithmetic sequence with the first term

=0.

A third sequence

1,

1,

2,

\ldots

is formed by adding corresponding terms of the two given sequences. The sum of the first ten terms of the third sequence is:

978

557

467

1068

not possible to determine from the information given

Write the two sequences as

a,ar,ar^2,\ldots

and

0,d,2d,\ldots

Use the first three sums and the condition

r\ne0

to determine

a,r,d

The first three termwise sums give

\begin{aligned} a&=1,\\ r+d&=1,\\ r^2+2d&=2. \end{aligned}

Substituting

d=1-r

yields

r(r-2)=0.

Since

r\ne0,

we have

r=2

and

d=-1.

The first ten geometric terms sum to

2^{10}-1=1023,

while the first ten arithmetic terms sum to

0-1-\cdots-9=-45.

Their combined sum is

1023-45=978.

Thus, the correct answer is A.

Problem 45 in other years