Question

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N.$ What is the smallest possible value of $N$ ?

$55$

$89$

$104$

$144$

$273$

Solution:

A sequence starting with $u,v$ has seventh term $5u+8v$ .

For two sequences $(a_1,a_2)$ and $(b_1,b_2)$ with different first terms, assume $a_1\lt b_1$ . Then $5a_1+8a_2=5b_1+8b_2$ , so $5(b_1-a_1)=8(a_2-b_2)$ .

Since $5$ and $8$ are relatively prime, $b_1-a_1$ is at least $8$ , and then nondecreasing order gives $b_2\ge b_1\ge a_1+8$ .

The smallest construction is $a_1=0$ , $b_1=b_2=8$ , and $a_2=13$ . This gives $N=5\cdot0+8\cdot13=104$ .

Thus, the correct answer is C .

2013 AMC 10B Problem 21

Solution:

Problem 21 in Other Years