Sum of a number of terms in g.p., Mathematics

We know that the terms in G.P. are:

a, ar, ar2, ar3, ar4, ................, arn-1

Let s be the sum of these terms, then

s = a + ar + ar2 + ar3 + ar4 + ................+ arn-1

                            or

s

 = 1058_GP.png

This is obtained as follows:

We know that

s = a + ar + ar2 + ar3 + ar4 +.............+ arn - 1                                            ......(1)

Multiplying this with "r" throughout, we have

r.s     = r.a + r.ar + r.ar2 + r.ar3 + r.ar4 +........+ r.arn-1 

         = ar + ar2 + ar3 + ar4 + ar5 +.......+ arn                                             ......(2)

Subtracting (1) from (2), we have

r.s - s = (ar - a) + (ar2 - ar) + (ar3 - ar2) +.......+ (arn - arn-1)

After canceling the terms equal in magnitude but opposite in sign, we are left with

s(r - 1) = arn - a
s(r - 1) = a(rn - 1)
 or s    = 427_GP1.png

By changing the signs in the numerator and the denominator we can also write the above equation as

 s  = 1058_GP.png

What happens to the above formula if the value of n is very large. The above formula can be written as

s

916_GP2.png

As the value of n approaches infinity (very large) the expression  2050_GP3.png becomes smaller to that extent where we ignore it. In this case the nth term is given as

Tn

= 1193_GP4.png

Now we look at a couple of examples.

Example

Find the sum of the series which is given below to 13 terms.

                   81, 54, 36, .............

The first term 'a' = 81 and the common ratio is obtained from the ratio of 54 and 81 or 36 and 54. It is 54/81 = 2/3. Now we employ the formula given above to calculate the sum of series to 13 terms.

s =   

1772_GP5.png

       = 241.78

The same series if considered as an infinite series, the sum of n terms would be

T = 491_GP6.png

= 243
Posted Date: 9/13/2012 4:36:13 AM | Location : United States







Related Discussions:- Sum of a number of terms in g.p., Assignment Help, Ask Question on Sum of a number of terms in g.p., Get Answer, Expert's Help, Sum of a number of terms in g.p. Discussions

Write discussion on Sum of a number of terms in g.p.
Your posts are moderated
Related Questions
Calculate the value of the following limits. Solution From the graph of this function illustrated below, We can illustrate that both of the one-sided limits suffer

The Invisible Effort :   Although the development of children is a process, what is noticed and given recognition to is the end-product. We usually speak of children having achieve


how to break fractions

The price of heating oil rose from $1.10 per gallon to $1.43 per gallon. What is the percent of increase? The price of heating oil rose $0.33 ($1.43 - $1.10 = $0.33). To ?nd ou

It is the last case that we require to take a look at. During this section we are going to look at solutions to the system, x?' = A x? Here the eigenvalues are repeated eigen

I am looking for a tutor in College Algebra

Determine the equation of the tangent line to r = 3 + 8 sinθ at θ = Π/6. Solution We'll first need the subsequent derivative. dr/dθ = 8 cosθ The formula for the deriv