Serious Series

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.

Answer.

See Serious Series for a solution.

(Update 1/18/2021, 9/6/2024) Other Solutions
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Tree Trunk Puzzle

Here is another problem (slightly edited) from the Sherlock Holmes puzzle book by Dr. Watson (aka Tim Dedopulos).

“Holmes and I were walking along a sleepy lane in Hookland, making our way back to the inn at which we had secured lodgings after scouting out the estates of the supposed major, C. L. Nolan. Up ahead, a team of horses were slowly pulling a chained tree trunk along the lane. Fortunately it had been trimmed of its branches, but it was still an imposing sight.

When we’d overtaken the thing, Holmes surprised me by turning sharply on his heel and walking back along the trunk. I stopped where I was to watch him. He continued at a steady pace until he’d passed the last of it, then reversed himself once more, and walked back to me.

‘Come along, old chap,’ he said as he walked past. Shaking my head, I duly followed.

‘It took me 140 paces to walk from the back of the tree to the front, and just twenty to walk from the front to the back,’ he declared.

‘Well of course,’ I said. ‘The tree was moving, after all.’

‘Precisely,’ he said. ‘My pace is one yard in length, so how long is that tree-trunk?’

Can you find the answer?”

Answer.

See the Tree Trunk Puzzle for solutions.

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The Triangle of Abū’l-Wafā’

I found an interesting geometric statement in a paper of Glen Van Brummelen cited in the online MAA January 2020 issue of Convergence:

“For instance, Abū’l-Wafā’ describes how to embed an equilateral triangle in a square, as follows: extend the base GD by an equal distance to E. Draw a quarter circle with centre G and radius GB; draw a half circle with centre D and radius DE. The two arcs cross at Z. Then draw an arc with centre E and radius EZ downward, to H. If you draw AT = GH and connect B, H, and T, you will have formed the equilateral triangle.”

So the challenge is to prove this statement regarding yet another fascinating appearance of an equilateral triangle.

See the Triangle of Abū’l-Wafā’

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Abraham Lincoln, Technologist

I thought there was nothing new we could learn about Abraham Lincoln, but I see I was quite mistaken after reading Sidney Blumenthal’s article,  “Abraham Lincoln, Tech Entrepreneur”.

In the current oppressive anti-science climate it is important to look back at our history and see how integral scientific thinking was to our founding and development.  Not only were our Founding Fathers scientists, such as Jefferson and Franklin who with others founded the American Philosophical Society in 1743, but it turns out that President Abraham Lincoln could also lay claim to a scientific mind.  Blumenthal’s article describes in detail how Lincoln employed science to advance the development of our country.  You should read the entire article, but I am including some highlights.

See Abraham Lincoln Technologist

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Quadrangle in Parallelogram

Here is another problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).

“A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle. Prove that at least one of the quadrangle’s diagonals is parallel to one of the parallelogram’s sides. (E. Sallinen)”

See Quadrangle in Parallelogram

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Three Counting Puzzles

Here are three counting puzzles from Alex Bellos’s book, Can You Solve My Problems?  Bellos recalls the famous legend of the young Gauss in the 19th century who summed up the whole numbers from 1 to 100 by finding a pattern that would simplify the work.  Bellos also mentioned that Alcuin some thousand years earlier had discovered a similar, but different, pattern to sum up the numbers.  In presenting these three problems he said, “The lesson … is this: If you’re asked to add up a whole bunch of numbers, don’t undertake the challenge literally.  Look for the pattern and use it to your advantage.”

Answer.

See Three Counting Puzzles for solutions.

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Amazing Radical Sum

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?”

Answer.

See Amazing Radical Sum for a solution.

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