MYTUTOR SUBJECT ANSWERS

354 views

What is the sum of the first n terms of a geometric sequence and where does it come from?

Recall that a geometric progression is a sequence (a_n) defined as follows

 

a_1 = a

a_n = a * r^(n - 1) for all integers n > 1

where a and r are some fixed numbers and r /= 0 is called the common ratio of a sequence.

 

Let's denote S_n as the sum of the first n terms of this sequence.

 

Case r = 1

Then all the terms are equal to a so

S_n = n*a

 

Case r /= 1

Then we have

S_n = a_1 + a_2 + … + a_n = a + a*r + … a*r^(n - 1) = a * (1 + r + … + r^(n – 1))

 

Now the formula for the difference between two n-th powers tells us that

r^n - 1 = r^n – 1^n = (r - 1) * (1 + r + … + r^(n-1))

and since r /= 1 we can divide both sides by r – 1 to have

1 + r + … + r ^(n - 1) = (r^n - 1) / (r – 1)

 

Finally, substituting this expression into the first equality we get

S_n = a * (r^n - 1) / (r – 1)

Kamil N. GCSE Maths tutor, A Level Maths tutor, 13 plus  Maths tutor,...

2 years ago

Answered by Kamil, an A Level Maths tutor with MyTutor


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

180 SUBJECT SPECIALISTS

£20 /hr

Jamie P.

Degree: Architecture (K100) (Bachelors) - Sheffield University

Subjects offered: Maths, Geography+ 1 more

Maths
Geography
Further Mathematics

“Hello! I am an Architecture student at The University of Sheffield. I would like to say I am creative, friendly and approachable, and willing to work hard in all aspects of what I do. My personable and easy-going nature will help to b...”

MyTutor guarantee

£22 /hr

Shivali J.

Degree: Medicine (Bachelors) - Imperial College London University

Subjects offered: Maths, Chemistry+ 5 more

Maths
Chemistry
Biology
.UKCAT.
.BMAT (BioMedical Admissions)
-Personal Statements-
-Medical School Preparation-

“Who am I? I am a first year medical student at Imperial College London. I have always been curious about the world we live in and have satisified this innate drive to understand the world around us through science. I hope to share thi...”

£20 /hr

Angus S.

Degree: Cognitive Science (BSc Hons) (Bachelors) - Edinburgh University

Subjects offered: Maths

Maths

“Hello! My name is Angus and I'm 19 years old. Currently, I'm a first year Informatics student at the University of Edinburgh. I spent part of a year after school travelling and teaching English as a foreign language, although Maths wa...”

MyTutor guarantee

About the author

Kamil N.

Currently unavailable: for new students

Degree: Mathematics and Computer Science (Masters) - Oxford, Merton College University

Subjects offered: Maths

Maths

“First year Maths & CompSci student from Oxford University, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

∫ log(x) dx

What is the importance of the Central Limit Theorem in carrying out hypothesis tests?

How do I find dy/dx for the following equation: (x^2) + 2y = 4(y^3) + lnx?

y = x*(x-2)^-1/2. Prove dy\dx = (x-4)/2*(x-2)^3/2

View A Level Maths tutors

Cookies:

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

mtw:mercury1:status:ok