MYTUTOR SUBJECT ANSWERS

627 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

305 SUBJECT SPECIALISTS

£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.”

Dylan  H. GCSE Further Mathematics  tutor, A Level Further Mathematic...
£26 /hr

Dylan H.

Degree: Materials Science and Engineering (Masters) - Imperial College London University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
Chemistry

“I offer tutoring in Maths, Chemistry and Physics and I am confident that I can help students to excel in exams through my engaging and interactive sessions. ”

£22 /hr

Sam T.

Degree: Mathematical Physics (Bachelors) - Edinburgh University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Chemistry

“I’m studying Physics with units in Maths and Chemistry. I really enjoy helping other people find a love for those subjects even if they think they're the worst.”

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

How do I rationalise the denominator of a fraction which consists of surds?

How do I find the coordinates of maximum and minimum turning points of a cubic equation?

Statistics: Dave throws a ball at a bucket. The probability the ball goes into the bucket is 0.4. Dave throws the ball four times. What is the probability that he gets it in twice?

Differentiate arctan(x) with respect to x. Leave your answer in terms of 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