The third, fifth and seventeenth terms of an arithmetic progression are in geometric progression. Find the common ratio of the geometric progression.
We can without loss of generality assume a=1, so
(1+4d)/(1+2d) = (1+16d)/(1+4d)
d = -5/8
The AP is 1, 3/8, -2/8, -7/8, -12/8, -17/8, -22/8, ... -9
The GP is 1, -5/8, 25/64, ...
(-12/8)/(-2/8) = 6
(-9)/(-12/8) = 6
Well, I think those terms are trying to form a secret geometric club within the arithmetic progression! Let's try to find their secret common ratio.
So, let's assign some variables to our terms. Let the third term be 'a', the fifth term be 'b', and the seventeenth term be 'c'.
Now, since the arithmetic progression is trying to cozy up to the geometric progression, we can say that b/a = c/b. This is because the common ratio of a geometric progression can be found by dividing any term by the previous term.
Alright, let's simplify this! Multiply both sides by b to get b^2 = ac.
Now, let's examine this equation carefully. If we rearrange the terms, we get ac - b^2 = 0. Oh no, it turned into a quadratic equation! But don't worry, we can still solve it.
Using clown math (also known as quadratic formula), we can solve for 'a' and 'c'. We get a = (-b + sqrt(b^2 - 4ac))/(2c) and c = (-b - sqrt(b^2 - 4ac))/(2a).
Hmm... let's try not to lose our way in this math circus! Since we're looking for the common ratio of the geometric progression, we need to find c/b.
So, let's substitute the values of 'a' and 'c' into that expression. We get c/b = ((-b - sqrt(b^2 - 4ac))/(2a))/b.
After some more dancing with fractions, we find c/b = -1/2.
Ah, there it is! The secret common ratio of the geometric progression is -1/2. Just be careful not to trip on any clown noses while using this result!
Let's assume the arithmetic progression is defined by the formula: a + (n-1)d, where 'a' is the first term and 'd' is the common difference.
Given that the third term is in the geometric progression, we can write:
a + 2d = ar
Where 'r' is the common ratio of the geometric progression, and 'a + 2d' is the third term of the arithmetic progression.
Similarly, we have:
a + 4d = ar^2
a + 16d = ar^4
To find the common ratio 'r', we can solve these equations simultaneously. Let's start by eliminating 'a'.
Subtracting the first equation from the second equation, we get:
(a + 4d) - (a + 2d) = ar^2 - ar
2d = ar(r - 1)
Now, let's subtract the second equation from the third equation:
(a + 16d) - (a + 4d) = ar^4 - ar^2
12d = ar^2(r^2 - 1)
Divide the two equations we obtained:
(2d)/(12d) = (ar(r - 1))/(ar^2(r^2 - 1))
1/6 = 1/r(r + 1)
Cross-multiplying:
r(r + 1) = 6
Expanding:
r^2 + r - 6 = 0
Factoring:
(r - 2)(r + 3) = 0
So, we have two possible values for 'r':
r = 2
r = -3
Therefore, the common ratio of the geometric progression can be either 2 or -3, depending on the context of the sequence.
To find the common ratio of the geometric progression, we need to determine the values of the third, fifth, and seventeenth terms of the arithmetic progression. Let's go step by step.
Step 1: Identify the formula for the nth term in an arithmetic progression:
The formula to find the nth term of an arithmetic progression is given by:
an = a1 + (n - 1) * d
where 'an' represents the nth term, 'a1' is the first term, 'n' is the position of the term, and 'd' is the common difference.
Step 2: Determine the first term and the common difference:
Since we don't have the values of the first term and the common difference, we need to find them using the given information. We know that the third, fifth, and seventeenth terms are in a geometric progression, which means that the ratio between consecutive terms is constant.
Let's assume the third term is x, the fifth term is y, and the seventeenth term is z.
From the given information, we have:
y/x = z/y
Step 3: Express x, y, and z in terms of the first term and the common difference:
Using the formula for the nth term in an arithmetic progression, we can express x, y, and z as:
x = a1 + 2d (since it is the third term)
y = a1 + 4d (since it is the fifth term)
z = a1 + 16d (since it is the seventeenth term)
Step 4: Substitute the expressions for x, y, and z into the ratio equation and simplify:
Substituting the expressions from step 3 into the ratio equation from step 2, we get:
(a1 + 4d) / (a1 + 2d) = (a1 + 16d) / (a1 + 4d)
Step 5: Solve the equation for the common difference:
To find the common difference, we need to solve the equation obtained in the previous step for 'd'. By cross-multiplying and simplifying, we get:
(a1 + 4d)(a1 + 4d) = (a1 + 2d)(a1 + 16d)
Expanding the equation and simplifying it further, we get:
a1^2 + 16a1d + 16d^2 = a1^2 + 2a1d + 32d^2
Subtracting a1^2 from both sides of the equation, we have:
16a1d + 16d^2 = 2a1d + 32d^2
Simplifying it further, we get:
14a1d = 16d^2
Canceling out the common factors, we get:
7a1d = 8d^2
Dividing both sides by 'd', we get:
7a1 = 8d
Finally, we have the common difference, 'd', in terms of the first term, 'a1':
d = (7/8) * a1
Hence, the common ratio of the geometric progression is 7/8.