MYTUTOR SUBJECT ANSWERS

740 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

417 SUBJECT SPECIALISTS

£26 /hr

Michelangelo M.

Degree: MMath Mathematics with Placement (Masters) - Bath University

Subjects offered:Maths, Italian+ 4 more

Maths
Italian
Further Mathematics
Computing
.STEP.
.MAT.

“Mathematics student at the University of Bath, willing to help you love this subject and improve your grades.”

£20 /hr

Rhianna C.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered:Maths, Chemistry+ 1 more

Maths
Chemistry
Biology

“I am currently studying Maths at University and absolutely love it! I can't wait to share my enthusiasm with you!”

£20 /hr

Sachin R.

Degree: Mathematics (Bachelors) - Bath University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“Hi, I’m Sachin and I study Maths at the University of Bath. I’m looking forward to working with my students to enable them to reach their highest potential.”

About the author

Matthew B.

Currently unavailable: for new students

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

Differentiate ((x^2)+1)^2

Find the shortest distance between the line L: x=1+t, y=1+2t, z=1-t and the point A: (2,3,4)

C1 - Simplifying a fraction that has a root on the denominator

What is the difference between a scalar and vector quantity?

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