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Given the geometric progression 625,125,25 find the sixth term and the sum of the 15 terms. How would i do this?

I am not understanding how to perform this operation... i know the formula is an=an-1+an-2

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Kirill Z. | Physics, math tutor with great knowledge and teaching skillsPhysics, math tutor with great knowledge...
4.9 4.9 (174 lesson ratings) (174)

Geometrical progression is given by the following formula:

an+1=an*q, where q is the common ratio. From this formula, you can obtain another one, relating the first term of progression to the arbitrary one,


So, for the sixth term in your case, we need to determine q and a1. a1=625; q=125/625, ratio of two successive terms, q=1/5.


Now, for the sum of N terms, from the first to the Nth one.


Let us multiply and divide this expression by (q-1). We will obtain the following:


Thus, for the sum of 15 terms we will get:



Hassan H. | Math Tutor (All Levels)Math Tutor (All Levels)
5.0 5.0 (176 lesson ratings) (176)

Hello Barbara,

The common ratio of a known geometric sequence can be found by

r = an+1/an

for some valid n.  In your case, of course, the common ratio turns out to be

r = 125/625 = 1/5.

Now, the formula you presented is not the correct formula for a geometric sequence.  (It is in fact the recursive definition of the Fibonacci  sequence.)  The explicit formula for the nth term of a geometric sequence is

an = a1rn-1.

Here, I am assuming that the sequence begins with index 1, i.e. a1 is the first term.

Since you know your first term (a1 = 625) and your common ratio (r = 1/5), you are now in position to compute the sixth term:

a6 = a1r6-1 = 625(1/5)5.

To find the sum of the first n terms of a geometric sequence, you use the following formula:

Sn = (a1(1-rn))/(1-r).

You can surely complete the computation on your own.


Hassan H.