The first three terms of a geometric progression are also the first ninth and eleventh terms respectic of an arithmetic progression.Given that the term of the geometric progression are all difference.find the common ratio(r),If the sum of infinity of the geometric progression is 8.find the first term common difference of the arithmetic progression
Can someone help me to solve it
a, ar, ar^2 are terms of an AP, so we know that
ar-a = 8d
ar^2 - ar = 2d
a/(1-r) = 8
solve for a,r,d and you find
GP: 6, 6/4, 6/16
AP: 96/16, 87/16, 78/16, 69/16, 60/16, 51/16, 42/16, 33/16, 24/16, 15/16, 6/16, ...
To find the common ratio (r) of the geometric progression, we can use the information provided.
Let's call the first term of the geometric progression "a". Then, the second term will be "ar" and the third term will be "ar^2" (since the terms have a common ratio of r).
According to the information given, the first term of the arithmetic progression (AP) corresponds to the first term of the geometric progression, which is "a". The ninth term of the arithmetic progression corresponds to the second term of the geometric progression, which is "ar", and the eleventh term corresponds to the third term of the geometric progression, which is "ar^2".
Therefore, we can write the following equations:
a = first term of AP
ar = ninth term of AP
ar^2 = eleventh term of AP
Since we have three equations and three unknowns (a, r, and the common difference of the arithmetic progression), we can solve for the common ratio (r) and the first term (a).
To find the common ratio (r), we can divide the equation for the ninth term by the equation for the first term:
(ar) / a = ninth term / first term
This gives us:
r = ninth term / first term
Substituting the values we know:
r = ar / a
r = "ar" / "a" (since ninth term = "ar" and first term = "a")
r = r
This equation tells us that the common ratio (r) is equal to itself. Since this is always true for any value of r, it does not provide a specific value for r.
Moving on to the second part of the question, to find the first term and the common difference of the arithmetic progression, we need additional information. The given sum of the infinite geometric progression is 8.
The sum of an infinite geometric progression is given by the formula:
S = a / (1 - r),
where S is the sum, a is the first term, and r is the common ratio.
Now we can substitute the given values into the sum formula:
8 = a / (1 - r)
From here, we would need to know a specific value for either the first term (a) or the common ratio (r) to solve for the other unknown. Without that information, we cannot find the first term or the common difference of the arithmetic progression.