What is the formula for the sum of a geometric sequence?

Question

12460 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-09-02T10:06:37

The formula is: #S_n=(a_1(1-q^n))/(1-q)#

To proove this formula you have first to write the sum of a geometric sequence:

#S_n=a_1+a_1*q+a_1*q^2+...a_1*q^(n-1)#

#S_n=a_1*(1+q+q^2+...q^n)#

We can multiply the last equality by #(1-q)#
We get:

#(1-q)*S_n=a_1*(1-q)*(1+q+q^2+...q^n)#

The right hand side can be written as #a_1*(1^n-q^n)# or
#a_1*(1-q^n)#. So finally we get:

#(1-q)*S_n=a_1*(1-q^n)#

Supposing #q!=1#, we can divide both sides by #1-q# and get:

#S_n=(a_1(1-q^n))/(1-q)#

That concludes the proof.