Resources tagged with: Summation of series

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Broad Topics > Patterns, Sequences and Structure > Summation of series

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.

The Root of the Problem

Age 14 to 18 Challenge Level:

Find the sum of this series of surds.

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.

Googol

Age 16 to 18 Short Challenge Level:

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

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.

Proof Sorter - Geometric Sequence

Age 16 to 18 Challenge Level:

Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?

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?

The Kth Sum of N Numbers

Age 16 to 18

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

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?

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.

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?

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?

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.

Double Trouble

Age 14 to 16 Challenge Level:

Simple additions can lead to intriguing results...

Speedy Summations

Age 16 to 18 Challenge Level:

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

An Introduction to Mathematical Induction

Age 16 to 18

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

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!

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 $S_1, S_2, ... S_{r-1}$.

A Swiss Sum

Age 16 to 18 Challenge Level:

Can you use the given image to say something about the sum of an infinite series?

Vanishing Point

Age 14 to 18 Challenge Level:

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

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?

Summing Squares

Age 14 to 16 Challenge Level:

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

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!

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 ?

Picture Story

Age 14 to 16 Challenge Level:

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

Slick Summing

Age 14 to 16 Challenge Level:

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

Sums of Powers - A Festive Story

Age 14 to 18

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

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?

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?

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?