the 5th, 9th and 16th term of an AP are consecutive terms of the GP.

Find.
1. the common ratio of the GP
2. given that the 13th term of the AP is 200, find the common difference of the AP.

To find the common ratio of the GP, we can use the formula for the nth term of a GP:

$$a_n = a_1 \cdot r^{n-1}$$

where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term.

1. The 5th, 9th, and 16th terms of the AP are consecutive terms of the GP. Let's represent the terms of the AP as \(a_5\), \(a_9\), and \(a_{16}\), respectively.

Based on the given information, we have:

$$a_9 = a_5 \cdot r^4$$
$$a_{16} = a_9 \cdot r^7$$

Since these are consecutive terms of the GP, we can set up the equation:

$$a_{16} = a_5 \cdot r^{15}$$

Substituting the expressions for \(a_9\) and \(a_{16}\), we get:

$$a_5 \cdot r^7 = a_5 \cdot r^4 \cdot r^{15}$$

We can cancel out \(a_5\) from both sides:

$$r^7 = r^4 \cdot r^{15}$$

Simplifying the equation:

$$r^{7} = r^{19}$$

Since the bases are the same, the exponents must be equal:

$$7 = 19$$

This equation is not true, so there is no constant common ratio for this GP.

2. If the 13th term of the AP is 200, we can use the formula for the nth term of an AP:

$$a_n = a_1 + (n - 1) \cdot d$$

where \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term.

Given that \(a_{13} = 200\), we can substitute the values into the formula:

$$200 = a_1 + (13 - 1) \cdot d$$
$$200 = a_1 + 12d$$

To find the common difference, we need another equation involving \(a_1\) and \(d\). We can use the information from the consecutive terms of the GP.

Using the formula for the nth term of the AP, we have:

$$a_5 = a_1 + (5 - 1) \cdot d$$
$$a_5 = a_1 + 4d$$

Since \(a_5\) is a term of the GP, it should be equal to \(a_1 \cdot r^4\):

$$a_1 + 4d = a_1 \cdot r^4$$

Now we have a system of equations:

$$\begin{cases}
200 = a_1 + 12d \\
a_1 + 4d = a_1 \cdot r^4
\end{cases}$$

We can solve this system of equations for \(a_1\) and \(d\).
Adding the equations, we get:

$$200 + a_1 + 4d + 12d = a_1 + 12d + a_1 \cdot r^4$$

Simplifying:

$$200 + 16d = 2a_1 + a_1 \cdot r^4$$

Now, we can substitute \(a_1 \cdot r^4\) using the equation from the GP:

$$200 + 16d = 2(a_1 + 4d)$$

Expanding:

$$200 + 16d = 2a_1 + 8d$$
$$200 + 8d = 2a_1$$

Now, we have an equation relating \(a_1\) and \(d\). However, we can't determine the exact values of \(a_1\) and \(d\) without additional information or another equation.

To find the common ratio of the GP, we need to determine the values of the 5th, 9th, and 16th terms of the arithmetic progression (AP).

1. The 5th term of an AP can be calculated using the formula:
5th term = First term + (5 - 1) * Common difference
= First term + 4 * Common difference

Similarly, the 9th term and 16th term can be found using:
9th term = First term + (9 - 1) * Common difference
= First term + 8 * Common difference

16th term = First term + (16 - 1) * Common difference
= First term + 15 * Common difference

2. Since the 5th, 9th, and 16th terms of the AP are consecutive terms of the geometric progression (GP), we can set up the following equation using the geometric progression formula:

(First term + 4 * Common difference) * Common ratio = (First term + 8 * Common difference)
(First term + 8 * Common difference) * Common ratio = (First term + 15 * Common difference)

We can simplify these equations:

(First term + 4 * Common difference) * Common ratio = First term + 8 * Common difference
First term * Common ratio + 4 * Common difference * Common ratio = First term + 8 * Common difference

(First term + 8 * Common difference) * Common ratio = First term + 15 * Common difference
First term * Common ratio + 8 * Common difference * Common ratio = First term + 15 * Common difference

We can subtract the first equation from the second equation to eliminate the First term * Common ratio term:

First term * Common ratio + 8 * Common difference * Common ratio - (First term * Common ratio + 4 * Common difference * Common ratio) = First term + 15 * Common difference - (First term + 8 * Common difference)

Doing the arithmetic, we get:
4 * Common difference * Common ratio = 7 * Common difference

We can divide both sides by Common difference:
4 * Common ratio = 7

Finally, dividing both sides by 4, we find:
Common ratio = 7/4 = 1.75

Therefore, the common ratio of the GP is 1.75.

To find the common difference of the AP, we can use the given information.

3. We know that the 13th term of the AP is 200. We can use the formula for finding the nth term of an AP:

13th term = First term + (13 - 1) * Common difference
= First term + 12 * Common difference

Substituting the known values, we have:
200 = First term + 12 * Common difference

From this equation, we can determine the common difference.

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