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There are **32** NRICH Mathematical resources connected to **Summation of series**, you may find related items under Patterns, sequences and structure.

Problem
Primary curriculum
Secondary curriculum
### Summing Geometric Progressions

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Slick Summing

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Harmonically

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picturing Square Numbers

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Clickety Click and All the Sixes

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picture Story

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Telescoping Series

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Swiss Sum

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Vanishing Point

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Googol

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

Age 16 to 18

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Seriesly

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

Age 16 to 18

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Clickety Click

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

Age 16 to 18

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Speedy Summations

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summing Squares

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Succession in Randomia

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Production Equation

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

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Sums of Powers - A Festive Story

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

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### An Introduction to Mathematical Induction

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

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### The Kth Sum of N Numbers

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

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Powerful Properties

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

Age 16 to 18

Interactive
Primary curriculum
Secondary curriculum
### Proof Sorter - Geometric Sequence

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

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Sum the Series

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.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Seriesly

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Reciprocal Triangles

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### More Polynomial Equations

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summats Clear

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Overarch 2

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### OK! Now Prove It

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

Age 16 to 18

Challenge Level