MYTUTOR SUBJECT ANSWERS

1099 views

How to prove that (from i=0 to n)Σi^2= (n/6)(n+1)(2n+1), by induction.

First you must show that the statement on the right hand side is true for n=1:

Σi=0 iwhen n=1, is equal to 12=1

(1/6)(1+1)(1+2)=(1/6)(2)(3)=1

This means that the statement is true for n=1.

Next you assume that it is true for 'k', where k is any number, and so you get;

Σi=0 i2 when n=k, is equal to (k/6)(k+1)(2k+2)

You then have to show that the statement is true for n=k+1 which would make;

Σi=0 i2 when n=k+1, is equal to (k+1)/6(k+2)(2k+3) call this 1)

As the left hand side is a sum, it can be written as;

Σi=0 i2 when n=k + (k+1)2

We already know the sum of i2 when n=k and so we can substitute it in;

(k/6)(k+1)(2k+1) + (k+1)2

We then try and reach 1)

We can factorise out (k+1)

(k+1)[(k/6)(2k+1) +k+1]

Next, multiply the inner brackets;

(k+1)[2k2/6+k/6 +k+1]

Take out a factor of 1/6

(k+1)/6(2k2+k+6k+6)= (k+1)/6(2k2+7k+6)

Finally, factorise the inner bracket;

(k+1)/6(k+2)(2k+3)

As this is equal to 1), we have proven that the statement is true for all values of n.

James B. A Level Maths tutor, GCSE Maths tutor

2 years ago

Answered by James, an A Level Maths tutor with MyTutor


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

253 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

Jacob F.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered:Maths, German+ 2 more

Maths
German
Further Mathematics
-Personal Statements-

“2nd year Maths undergrad with enhanced DBS (CRB) certification and experience teaching in secondary schools and at University. Fluent German speaker.”

£26 /hr

Joe C.

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“I'm a third year student at Bristol University studying a Masters Degree in Mathematics. ”

About the author

£26 /hr

James B.

Degree: Mathematics (Masters) - Exeter University

Subjects offered:Maths

Maths

“About MeI'm a second year student at Exeter University. I've been passionate about maths from a young age and have carried that love through my school and uni career. I hope that I can generate the same passion for my tutees as well....”

You may also like...

Posts by James

How do you differentiate X to the power of a?

How to prove that (from i=0 to n)Σi^2= (n/6)(n+1)(2n+1), by induction.

What is "Standard Form"?

What is a complex number?

Other A Level Maths questions

Differentiate y = (6x-13)^3 with respect to x

How to solve polynomials

The point P lies on the curve C: y=f(x) where f(x)=x^3-2x^2+6x-12 and has x coordinate 1. Find the equation of the line normal to C which passes through P.

Where do the kinematics equations (SUVAT) come from?

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