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Advanced Problem Solving Module 23

Sequences and Series
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Sequences and Series

Challenge level
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Sequences are everywhere in mathematics! In these problems you can explore some sequences, learn how to represent them, and how to calculate sums of series.
Summats Clear
problem

Summats Clear

Age
16 to 18
Challenge level
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Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.
Be reasonable
problem

Be reasonable

Age
16 to 18
Challenge level
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Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.
Harmonically
problem
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Harmonically

Age
16 to 18
Challenge level
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Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?