Here is another typical sum puzzle from Presh Talwalkar.
“Solve the following sums:
_____1/(1×3) + 1/(3×5) + 1/(5×7) + 1/(7×9) + 1/(9×11) =
_____1/(4×7) + 1/(7×10) + 1/(10×13) + 1/(13×16) =
_____1/(2×7) + 1/(7×12) + 1/(12×17) + … =”
The only reason I am including this puzzle is that Talwalkar gets very excited about deriving a formula that can solve sums of this type. This gives me an opportunity to discuss the “formula vs. procedure” way of doing math.
See the Incredible Trick Puzzle
This is another stimulating little problem from the 2022 Math Calendar.
“a1 = 1, a2 = 2, …, an+1 = an + 6an-1
x = lim an+1/an as n → ∞
Solve for x.”
As before, recall that all the answers are integer days of the month.
See Stimulating Sequence
This is another infinite series from Presh Talwalkar, but with a twist.
“This problem is adapted from one given in an annual national math competition exam in France. Evaluate the infinite series:
1/2! + 2/3! + 3/4! + …”
The twist is that Talwalkar provides three solutions, illustrating three different techniques that I in fact have used before in series and sequence problems. But this time I actually found a simpler solution that avoids all these. You also need to remember what a factorial is: n! =n(n – 1)(n – 2)…3·2·1.
See a Nice Factorial Sum
This is a belated Christmas puzzle from December 2019 MathsMonday.
“A Christmas tree is made by stacking successively smaller cones. The largest cone has a base of radius 1 unit and a height of 2 units. Each smaller cone has a radius 3/4 of the previous cone and a height 3/4 of the previous cone. Its base overlaps the previous cone, sitting at a height 3/4 of the way up the previous cone.
What are the dimensions of the smallest cone, by volume, that will contain the whole tree for any number of cones?”
Recall that the volume of a cone is π r2 h/3.
See Another Christmas Tree Puzzle
Here is another sum problem, this time from the 2021 Math Calendar.
As before, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.
See the Winter Sum
Here is another problem from the 2020 Math Calendar.
As a hint, recall that all the answers are integer days of the month. And the solution employs a technique familiar to these pages.
See Autumn Sum
The following problem comes from a 1961 exam set collected by Ed Barbeau of the University of Toronto. The discontinued exams (by 2003) were for 5th year Ontario high school students seeking entrance and scholarships for the second year at a university.
“If sn denotes the sum of the first n natural numbers, find the sum of the infinite series
Unfortunately, the “Grade XIII” exam problem sets were not provided with answers, so I have no confirmation for my result. There may be a cunning way to manipulate the series to get a solution, but I could not see it off-hand. So I employed my tried and true power series approach to get my answer. It turned out to be power series manipulations on steroids, so there must be a simpler solution that does not use calculus. I assume the exams were timed exams, so I am not sure how a harried student could come up with a quick solution. I would appreciate any insights into this.
See Serious Series
(Update 1/18/2021) Another Solution Continue reading
The craziness of manipulating radicals strikes again. This 2006 four-star problem from Colin Hughes at Maths Challenge is really astonishing, though it takes the right key to unlock it.
“Problem Consider the following sequence:
For which values of [positive integer] n is S(n) rational?”
See Amazing Radical Sum.
This is a delightful and surprising problem from Presh Talwalkar.
“This puzzle was created by a MindYourDecisions fan in India. What is the value of the infinite product? The numerators are the odd nth roots of [Euler’s constant] e and the denominators are even nth roots of e.”
See Euler Magic
If you will pardon the pun, this is a diabolical problem from the collection Five Hundred Mathematical Challenges.
“Problem 5. Calculate the sum
It has a non-calculus solution, but that involves a bunch of manipulations that were not that evident to me, or at least I doubt if I could have come up with them. I was able to reframe the problem using one of my favorite approaches, power series (or polynomials). The calculations are a bit hairy in any case, but I was impressed that my method worked at all.
See the Number of the Beast