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Broad Topics > Advanced Algebra > Summation of series

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Harmonically

Age 16 to 18 Challenge Level:

Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?

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OK! Now Prove It

Age 16 to 18 Challenge Level:

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

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Summats Clear

Age 16 to 18 Challenge Level:

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.

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Proof Sorter - Geometric Series

Age 16 to 18 Challenge Level:

This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.

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Production Equation

Age 16 to 18 Challenge Level:

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

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2^n -n Numbers

Age 16 to 18

Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.

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Overarch 2

Age 16 to 18 Challenge Level:

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

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Summing Geometric Progressions

Age 14 to 18 Challenge Level:

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

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The Kth Sum of N Numbers

Age 16 to 18

Yatir from Israel describes his method for summing a series of triangle numbers.

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Weekly Challenge 36: Seriesly

Age 16 to 18 Short Challenge Level:

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

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Sums of Powers - A Festive Story

Age 11 to 16

A story for students about adding powers of integers - with a festive twist.

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Vanishing Point

Age 14 to 18 Challenge Level:

How can visual patterns be used to prove sums of series?

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Slick Summing

Age 14 to 16 Challenge Level:

Watch the video to see how Charlie works out the sum. Can you adapt his method?

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Weekly Challenge 34: Googol

Age 16 to 18 Short Challenge Level:

Find the smallest value for which a particular sequence is greater than a googol.

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Speedy Summations

Age 16 to 18 Challenge Level:

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

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Double Trouble

Age 14 to 16 Challenge Level:

Simple additions can lead to intriguing results...

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Clickety Click

Age 16 to 18 Short Challenge Level:

What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

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An Introduction to Mathematical Induction

Age 16 to 18

This article gives an introduction to mathematical induction, a powerful method of mathematical proof.

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Picture Story

Age 11 to 16 Challenge Level:

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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More Polynomial Equations

Age 16 to 18 Challenge Level:

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

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Degree Ceremony

Age 16 to 18 Challenge Level:

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

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Clickety Click and All the Sixes

Age 16 to 18 Challenge Level:

What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

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Sum the Series

Age 16 to 18

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.

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Seriesly

Age 16 to 18 Challenge Level:

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

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Reciprocal Triangles

Age 16 to 18 Challenge Level:

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

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Telescoping Series

Age 16 to 18 Challenge Level:

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

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Succession in Randomia

Age 16 to 18 Challenge Level:

By tossing a coin one of three princes is chosen to be the next King of Randomia. Does each prince have an equal chance of taking the throne?