By Martin Erickson
Every mathematician (beginner, beginner, alike) thrills to discover easy, stylish strategies to possible tough difficulties. Such satisfied resolutions are known as ``aha! solutions,'' a word popularized by means of arithmetic and technological know-how author Martin Gardner. Aha! ideas are extraordinary, gorgeous, and scintillating: they demonstrate the great thing about mathematics.
This publication is a suite of issues of aha! ideas. the issues are on the point of the varsity arithmetic pupil, yet there will be anything of curiosity for the highschool scholar, the trainer of arithmetic, the ``math fan,'' and somebody else who loves mathematical challenges.
This assortment comprises 100 difficulties within the parts of mathematics, geometry, algebra, calculus, likelihood, quantity concept, and combinatorics. the issues start off effortless and usually get more challenging as you move throughout the e-book. a number of options require using a working laptop or computer. a big characteristic of the publication is the bonus dialogue of similar arithmetic that follows the answer of every challenge. This fabric is there to entertain and tell you or aspect you to new questions. when you do not take into accout a mathematical definition or thought, there's a Toolkit at the back of the e-book that might help.
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Additional info for Aha! Solutions
We see fractions with 11 in the denominator in both cases. Reflecting on this, we realize that in the course of twelve hours the minute hand and hour hand coincide eleven times. Specifically, this happens at 12:00; at some time between 1:00 and 2:00; between 2:00 and 3:00 (we already figured this one out); between 3:00 and 4:00 (we figured this one out, too); between 4:00 and 5:00; between 5:00 and 6:00; between 41 42 2 Intermediate Problems 6:00 and 7:00; between 7:00 and 8:00; between 8:00 and 9:00; between 9:00 and 10:00; and between 10:00 and II :00.
If n is odd, then v is an integer, since it is equal to the middle term of the sequence. However, if n is even, then v is not an integer; it is equal to the average of the two middle terms of the sequence (and hence is a half-integer). So it seems that we must consider the cases n odd and n even separately, and that the n even case is a little tricky. Believe it or not, there is a way to ensure that n is odd, and this is where the aha! realization comes in. The problem asks for sequences of positive integers but let's relax the condition temporarily to any integers (positive, negative, or zero).
4 No Calculus Needed 39 Our inequality is established, with equality if and only if al = a2 = a3 = a4. We can use the same procedure to prove Jensen's inequality for n any power of 2. But how do we prove it for, say n = 3? We use a little technique credited to Augustin Louis Cauchy (1789-1857). Set a4 = (al + a2 + a3)/3. Then (al + a2 + a3 + a4)/4 = (al + a2 + a3)/3, and hence ! (a3) + ! 4 (a +al+a l 3 ) . Simplifying, we obtain = = a2 a3. The general statement of Jensen's inequality Equality occurs if and only if al can be proved similarly or by induction on n.