Formal Series
A is the set of natural numbers, greater than or equal to 2 . Also, the set of pairs B = { ( a , b ) ∣ a , b ∈ A , a < b } is defined.
there is an infimum to the set below and we are interested in finding the pair ( a , b ) ∈ B for which the infimum of the set below is achieved.
{ ∑ i = 1 ∞ ∑ j = 1 ∞ ( a 1 ) i ( b 1 ) j ( − 1 ) i + j 1 = 2 k ∣ ∣ ∣ ∣ k ∈ N , ( a , b ) ∈ B }
enter the final solution as a × b
The answer is 21.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
gut gemacht! :)
Good work! (And, yes, I know what the German above means.)
Log in to reply
Awesome! I suppose, you knew without "google translate"s aid :D
Log in to reply
Yes, this case, it was part of the tiny amount of German language that I have left from my infancy, 7 decades ago. My great grandmother, who babysat me while my parents worked, was much more fluent in German than in English, so that was the language she used used during the day. I also studied Latin in high school and Russian in college.
Considering geometrical series, we find that ∑ i = 1 ∞ ∑ j = 1 ∞ ( a 1 ) i ( b 1 ) j ( − 1 ) i + j 1 = ( a + 1 ) ( b + 1 ) . It is required that both a + 1 and b + 1 be powers of 2. With the given constraints, 1 < a < b , the minimum, k = 5 , is attained when ( a , b ) = ( 3 , 7 ) , and the answer is 2 1 .