The second, third and ninth terms of an arithmetic progression forms the first, second and third terms of a geometric progression. Find the common ratio and hence the eleventh term of geometric progression.
Answer of the common ratio =6
To find the common ratio of the geometric progression, we need to first find the common difference of the arithmetic progression.
Let's assume the first term of the arithmetic progression is "a" and the common difference is "d".
The second term of the arithmetic progression is "a + d" (which is also the first term of the geometric progression).
The third term of the arithmetic progression is "a + 2d" (which is also the second term of the geometric progression).
The ninth term of the arithmetic progression is "a + 8d" (which is also the third term of the geometric progression).
Since the second, third, and ninth terms of the arithmetic progression form a geometric progression, we can set up the proportion:
(a + d) / a = (a + 2d) / (a + d) = (a + 8d) / (a + 2d)
Now, let's solve for the common difference "d":
(a + d) / a = (a + 2d) / (a + d)
Cross-multiplying, we get:
(a + d)(a + d) = a(a + 2d)
Expanding and simplifying:
a^2 + 2ad + d^2 = a^2 + 2ad
Canceling out the common terms:
d^2 = 0
Since d^2 = 0, we can conclude that the common difference "d" is 0. Therefore, the arithmetic progression is actually a sequence of the same number repeated.
Now, let's find the common ratio of the geometric progression using the first term of the arithmetic progression (which is also the first term of the geometric progression):
Common ratio (r) = (a + d) / a = (a + 0) / a = 1
Since the common ratio is 1, the geometric progression is also a sequence of the same number repeated.
To find the eleventh term of the geometric progression, we can use the formula:
nth term = first term * (common ratio)^(n-1)
where n is the position of the term in the sequence.
In this case, since the common ratio is 1, each term of the geometric progression is the same as the first term. Thus, the eleventh term is equal to the first term, which is the same as any other term in the sequence.