This course begins with sequences and series and then studies the progressions that follow a fixed rule. For arithmetic progressions it covers the general term, the sum of n terms, six useful properties, a smart way to choose unknown terms, arithmetic means and a set of word problems. Geometric progressions follow the same path, adding the sum to infinity, recurring decimals and the geometric mean, with the proof that the arithmetic mean is never less than the geometric mean. The course ends with arithmetico-geometric series, summed by the multiply and subtract method, and harmonic progressions, where all three means are compared.
Middle Years
Progressions
Sequences and series, arithmetic and geometric progressions with their general terms, sums, properties and means, arithmetico-geometric series, and harmonic progressions with the harmonic mean.
- 13 lessons
Overview
Lessons
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Sequences and series
What a sequence and a series are, the general term, recursive rules, finite and infinite sequences, testing whether a number is a term, and finding terms from the sum.
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Arithmetic progressions and the general term
The common difference, the general term a + (n − 1)d, finding which term a number is, counting terms, terms from the end, and recognising an A.P. from its formula.
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The sum of n terms of an A.P.
Gauss's forwards and backwards trick, the two forms of the sum formula, the sums of natural, odd and even numbers, and finding the general term from the sum.
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Properties of an A.P. and selecting terms
Six properties of arithmetic progressions, including 2b = a + c and equal sums from the two ends, and how to choose three, four or five unknown terms.
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Arithmetic means
The arithmetic mean of two or more numbers, inserting any number of arithmetic means between two numbers, and the sum of the inserted means.
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Problems on arithmetic progressions
Word problems on savings, seating, loan instalments and stacked logs, plus algebraic problems that use the term and sum formulas together.
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Geometric progressions and the general term
The common ratio, the general term arⁿ⁻¹, finding which term a number is, terms from the end, and growth or decay by a fixed percentage.
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The sum of a geometric progression
The sum of n terms by multiplying by r and subtracting, the sum to infinity when the ratio lies between −1 and 1, recurring decimals and a bouncing ball.
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Properties of a G.P. and selecting terms
Five properties of geometric progressions, including b² = ac and equal products from the two ends, logarithms of a G.P., and choosing unknown terms.
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Geometric means
The geometric mean √(ab), inserting geometric means, the product of the inserted means, the proof that A.M. ≥ G.M., and finding two numbers from their means.
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Problems on geometric progressions
Problems on doubling, compound interest and depreciation, series such as 7 + 77 + 777, recurring decimals, and sums of a G.P. and of its squares.
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Arithmetico-geometric progressions
Series whose terms multiply an A.P. by a G.P.: the general term, the sum of n terms by the multiply and subtract method, and the sum to infinity.
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Harmonic progressions and the harmonic mean
Progressions whose reciprocals form an A.P., their general term and properties, the harmonic mean, inserting harmonic means, and A.M. ≥ G.M. ≥ H.M.