MYTUTOR SUBJECT ANSWERS

483 views

How do I add up the integers from 1 to 1000 without going insane?

Suppose you are asked by your teacher to sum all the integers from 1 up to 1000. You might be thinking they must hold some kind of grudge against you. If you try to calculate the sum by adding on each integer one at a time, you will have to perform 999 separate additions, some of which will be quite long and tedious. This would take far longer than anyone can be bothered to spend adding up numbers.

There is, however, a quicker way. Let's give our sum a letter to represent its unknown value. Let's use "S" for "sum". Then:

S = 1 + 2 + 3 + ... + 999 + 1000

But we can rewrite this sum in reverse order. Starting with 1000 and ending with 1:

S = 1000 + 999 + 998 + ... + 2 + 1

We can now add together these two equations to give us:

S + S = (1+1000) + (2+999) + (3+998) + ... + (999+2) + (1000+1)

Simplifying both sides gives us:

2S = 1001 + 1001 + 1001 + ... + 1001 + 1001

The right hand side has 1000 separate terms, since our original sum contained 1000 numbers. So:

2S = 1000 x 1001 = 1001000

Dividing both sides by two we find that S = 500500. Therefore:

1 + 2 + 3 + ... + 999 + 1000 = 500500

This calculation is an example of a more general concept called an arithmetic series, where you sum a sequence of numbers which differ by adding on a fixed amount with each step.

Matthew B. A Level Maths tutor, A Level Further Mathematics  tutor, A...

2 years ago

Answered by Matthew, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

291 SUBJECT SPECIALISTS

£20 /hr

Iebad A.

Degree: Aerospace Engineering (Masters) - Sheffield University

Subjects offered:Maths, Italian

Maths
Italian

“About me: Hello, my name is Iebad and I am an aerospace engineering student at The University of Sheffield. As an engineer, maths is the key to unlock all the solutions to our problems in this field. Since the early years I had a broad...”

£20 /hr

Joshua D.

Degree: Economics and Mathematics (Bachelors) - Bath University

Subjects offered:Maths, Science+ 2 more

Maths
Science
Physics
Economics

“Enthusiastic tutor from the University of Bath, aiming to push pupils to achieve the best that they can.”

MyTutor guarantee

£26 /hr

Priya L.

Degree: Economics (Bachelors) - Warwick University

Subjects offered:Maths, Further Mathematics + 1 more

Maths
Further Mathematics
Economics

“About Me: I recently graduated from the University of Warwick with an Economics degree. I am currently on a gap year before I begin my graduate role as a Management Consultant in October. I decided to tutor because I wanted to spend m...”

About the author

£22 /hr

Matthew B.

Degree: Mathematics (Masters) - Warwick University

Subjects offered:Maths, Science+ 2 more

Maths
Science
Physics
Further Mathematics

“About me: Hello! My name is Matt and I'm a 3rd year Maths student at the University of Warwick. I achieved all A*s at GCSE and 4 A*s at A-Level in Maths, Further Maths, Physics, and Chemistry. I am offering tutoring forA-Level Maths, ...”

You may also like...

Other A Level Maths questions

Find the general solution, in degrees, of the equation 2sin(3x+45°)=1. Use your general solution to find the solution of 2sin(3x+45°)=1 that is closest to 200 °.

Solve the equation; 4 cos^2 (x) + 7 sin (x) – 7 = 0, giving all answers between 0° and 360°.

How do you add or subtract complex numbers?

How do I integrate terms with sin^2(x) and cos^2(x) in them? For example integrate (1+sin(x))^2 with respect to x

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok