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

Answer: A

Concepts:geometric sequence arithmetic sequence systems of equations

Difficulty rating: 1930

Small Hint:

Write the two sequences as $a,ar,ar^2,\ldots$ and $0,d,2d,\ldots$

Big Hint:

Use the first three sums and the condition $r\ne0$ to determine $a,r,d$

Solution:

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.

1955 AMC 12 Problem 45

Answer: A

Difficulty rating: 1930

Small Hint:

Big Hint:

Solution:

Problem 45 in Other Years