Problem:
Let Sn be the sum of the first n term of an arithmetic sequence that has a common difference of 2. The quotient SnS3n does not depend on n. What is S20?
Answer Choices:
A. 340
B. 360
C. 380
D. 400
E. 420
Solution:
Let a be the first term of the arithmetic sequence. The sum of the first n terms of an arithmetic sequence with common difference 2 is
Sn=2n(a+(a+(n−1)⋅2))=n(a+n−1).
Because SnS3n does not depend on n, it follows that S1S3=S2S6. This means that
a3(a+2)=2(a+1)6(a+5).
Solving this equation gives a=1.
Indeed, if the first term of an arithmetic sequence is 1 and its common difference is 2, then
SnS3n=n(1+n−1)3n(1+3n−1)=9
which does not depend on n. Therefore there is one and only one arithmetic sequence that satisfies the given properties, namely 1,3,5,… Thus S20=20(1+20−1)=(D)400. In fact, Sn=n2.
OR
As in the first solution, the sum of an arithmetic sequence with first term a and common difference 2 is Sn=n(n+a−1). Therefore
SnS3n=n(n+a−1)(3n)(3n+a−1)=3⋅n+a−13n+a−1=9⋅1+na−11+3na−1.
This value approaches 9 as n→∞, so if SnS3n does not depend on n, then it must be identically 9 and a must equal 1. Therefore S20=20⋅(20+1−1)=(D)400.
The problems on this page are the property of the MAA's American Mathematics Competitions