Level 7: Series and sigma

Patterns and sequences · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

In level 6 you added up a geometric sequence with one clever subtraction. In this level adding up becomes the main subject. You learn the sigma shorthand for long sums, derive the arithmetic series formula carefully, and add infinitely many numbers when the terms shrink fast enough. Then you meet the golden ratio hiding in the Fibonacci numbers, prove formulas for sums of squares and cubes, and finish by seeing every sequence as a function whose graph is a row of dots on a line, a parabola or an exponential curve.

Lesson	You will be able to
Sigma notation	Read and write sums with \(\Sigma\), count their terms, change the starting index, and split and simplify sums
Arithmetic series, carefully	Derive \(S_n = \frac{n(a + l)}{2}\), and add terms that meet a condition, such as the multiples of 7 between 100 and 500
Infinite geometric series	Explain why \(a + ar + ar^2 + \cdots = \frac{a}{1 - r}\) when \(\lvert r \rvert < 1\), spot when there is no sum, and find a bouncing ball's total distance
Recursion and the golden ratio	Show why Fibonacci ratios approach \(\varphi = \frac{1 + \sqrt5}{2}\) from \(x^2 = x + 1\), and use and check Binet's formula
Sums of squares and cubes	Find and prove \(\sum k^2 = \frac{n(n+1)(2n+1)}{6}\) and \(\sum k^3 = \left(\frac{n(n+1)}{2}\right)^2\), and use them
Sequences as functions	Treat a sequence as a discrete function, link arithmetic, quadratic and geometric sequences to lines, parabolas and exponentials, and describe convergence

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