Problem:
What is the value of
(k=1∑20log5k3k2)⋅(k=1∑100log9k25k)?
Answer Choices:
A. 21
B. 100log53
C. 200log35
D. 2,200
E. 21,000
Solution:
Note that
log5k3k2=log5klog3k2=klog5k2log3=log5log3⋅k
and
log9k25k=log9klog25k=2klog32klog5=log3log5.
Thus
k=1∑20log5k3k2=log5log3k=1∑20k=log5log3⋅220⋅21=210⋅log5log3
and
k=1∑100log9k25k=100⋅log3log5
The requested product is 210⋅100=21,000.
The problems on this page are the property of the MAA's American Mathematics Competitions