# the first three terms of a GP are the first, fourth and tenth terms of an AP. given that the first term is 6 and that all the terms of a GP are different, find the common ratio.

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## a = 6

ar = a+3d
ar^2 = a+9d
so,
d = (ar-a)/3

ar^2 = a+9a(r-1)/3
r^2 = 1+3r-3
r^2-3r+2 = 0
r = 1 or 2

since the GP terms are different, r=2 and d=2 and the sequences are

6,12,24
6,8,10,12,14,16,18,20,22,24

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## Let's assume that the first term of the GP is a, and the common ratio is r.

The first term of the AP is 6, the fourth term is a, and the tenth term is 6r^6.
We can find the common difference of the AP using the formula a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the position of the term, and d is the common difference.

Using this formula, we have:
6 = 6 + (4-1)d => 6 = 6 + 3d => 3d = 0 => d = 0

Since the common difference of the AP is zero, all the terms of the AP are the same.

Since the first term of the AP is 6 and all the terms are the same, we can conclude that the first term of the GP is equal to 6.

So, a = 6

Now, we can find the common ratio of the GP.
The fourth term of the GP is ar^3 = a = 6.
So, 6r^3 = 6.

Dividing both sides of the equation by 6, we get:
r^3 = 1

Taking the cube root of both sides of the equation, we get:
r = 1

Therefore, the common ratio of the GP is 1.

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## To find the common ratio in a geometric progression (GP), we need to use the information given about the terms of the AP.

Let's proceed step by step:

Step 1: Write out the terms of the AP.
The first, fourth, and tenth terms of the AP are given. Let's label them as a, d, and t respectively.
First term (a) = 6
Fourth term (d) = ?
Tenth term (t) = ?

Step 2: Find the common difference of the AP.
To find the common difference (d), we can use the formula:
d = (t - a) / (10 - 1)
Substituting the known values:
d = (t - 6) / 9

Step 3: Find the fourth term of the AP.
We can find the fourth term (d) using the formula:
d = a + 3d
Substituting the known values:
6 + 3d = d
2d = 6
d = 3

Step 4: Find the tenth term of the AP.
Using the formula from Step 2, we can solve for the tenth term (t):
3 = (t - 6) / 9
27 = t - 6
t = 33

Step 5: Find the common ratio of the GP.
Since the first term of the GP is 6 and the fourth term is 3, we can use the formula:
common ratio = fourth term / first term
common ratio = 3 / 6
common ratio = 1/2

Therefore, the common ratio in this geometric progression (GP) is 1/2.