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
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
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.
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.