3+6+9+ …………………+3n=(3n(n+1))/2

Expert Answers

An illustration of the letter 'A' in a speech bubbles

I assume you are asked to show that the finite sum (this is a series; the sum of the elements of some sequence):


We can rewrite the equation as :


Which is equivalent to 1+2+3+...+n=n(n+1)/2.

The nth partial sums are called the triangular numbers. 1,1+2=3,1+2+3=6,1+2+3+4=10, etc...

Each of 1, 3, 6, 10, 15, 21,... can be drawn as a set of dots in an equilateral triangle (1 being a degenerate case.)

(1) For each partial sum we can draw an nx(n+1) rectangle. (Note that one of n or (n+1) will be even, so the product n(n+1) is even.) Drawing a diagonal line so that there are the same number of dots on each side yields "L" shaped figures with 1, 3, 6, 10, etc... dots.






Etc. These triangles have 1, 1+2, 1+2+3, etc. dots in them and can be seen to be 1/2 the number of dots in a nx(n+1) rectangle. So 1+2+3+...+n =(n(n+1))/2.

So 1+2+3+...+n=(n(n+1))/2 and 3+6+9+...+3n=(3n(n+1))/2 as required.

(2) A possible apocryphal story about the mathematician Karl F. Gauss has the young Gauss in a class when the teacher asks the students to add up the numbers from 1 to 100 (perhaps hoping for some quiet time.) Young Gauss almost immediately has the answer. His method:

Write the sum in one direction. Beneath write the sum in reverse order and add the two series.



There are 100 101's, which is 10100. But that is adding the series twice, so divide by 2 to get 5050, which is the answer.

Thus, for a finite arithmetic series (each term in the underlying sequence differs from the previous term by the same common difference), the sum can be found by:

`S_(n)=(n(a_1+a_n))/2` where n is the number of terms, a(1) is the first term and a(n) is the nth term. For Gauss, we have `S_(100)=(100(1+100))/2`

For the arithmetic series 3+6+9+...+3n, we get:

`S_(n)=(n(3+3n))/2=(3n(1+n))/2` as required.

Approved by eNotes Editorial Team

Posted on

Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial