geometric progression how to determine the sum of the first

geometric progression

how to determine the sum of the first term and the common ration of a g p if we know other terms?

Solution

I\'ll give you one example.

We\'ll write the formula for the general term of any geometric progression:

bn=b1*r^(n-1), where b1 is the first term, r is the common ratio

Since we know the values of the 6th and the 8th terms, from enunciation, we\'ll substitute them into the formula of the general terms:

b8=b7*r

But b7=b6*r

So, b8=b6*r*r=b6*r^2

9=25*r^2

We\'ll divide by 25 both sides:

r = sqrt (9/25)

r = 3/5

But the 6th term could be written as:

b6=b1*r^5

25=b1*(3/5)^5

b1=5^7/3^5

b1+r=(5^7/3^5) + (3/5)

b1+r=(5^8 + 3^6)/5*3^5