The ratio of the fourth and 12th term of a g.p. With positive common ratio is 1:256. If the difference of two terms be 61.2, find the sum of 8 terms of this series.
ar^3/ar^11 = 1/r^8 = 1/2^8
so, r = 2
I expect you can take it from there...
This might help you to learn about a.p. and g.p.:
http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-apgp-2009-1.pdf
Thanks
I don't trust anyone who tries to impersonate me and can't even get my name right.
To solve this problem, we need to find the values of the first term (a) and the common ratio (r) of the geometric progression (g.p.).
Let's start by using the given information: the ratio of the fourth and 12th terms is 1:256.
The formula for finding the nth term of a geometric progression is given by:
an = a * r^(n-1)
So, for the fourth term (a4) and the 12th term (a12), we have:
a4 = a * r^3
a12 = a * r^11
Given that the ratio of a4 to a12 is 1:256, we can set up the following equation:
a4/a12 = 1/256
Now let's substitute the expressions for a4 and a12:
(a * r^3)/(a * r^11) = 1/256
We can simplify this equation by canceling out the "a" terms:
r^(11-3) = 256
Simplifying further:
r^8 = 256
To solve for "r," take the eighth root of both sides:
r = 2
Now that we have the value of "r," we can find the value of "a" by using the difference between two terms, which is given as 61.2.
The formula for the difference between two terms in a geometric progression is:
difference = a * (1 - r^n) / (1 - r)
Let's use this formula to solve for "a":
61.2 = a * (1 - 2^3) / (1 - 2)
Simplifying:
61.2 = a * (1 - 8) / (-1)
61.2 = -7a
Dividing both sides by -7:
a = -8.74
Now that we have the values for "a" and "r," we can find the sum of eight terms of the geometric progression.
The formula for the sum of n terms of a geometric progression is:
Sn = a * (r^n - 1) / (r - 1)
Substituting the values of "a" and "r" into the formula:
S8 = (-8.74 * (2^8 - 1)) / (2 - 1)
Simplifying:
S8 = (-8.74 * (256 - 1)) / 1
S8 = (-8.74 * 255) / 1
S8 = -2230.7
Therefore, the sum of eight terms of this geometric progression is approximately -2230.7.