Determine the first three terms of the geometric progression of which the 7th term is 192 and te second term is 6

ar^6 = 192

ar = 6

now just divide, and you get

r^6 = 64

now take it from there.

2(8dived by2)exponent2 times 2equals to 64

To determine the first three terms of the geometric progression, we need to find the common ratio and use it to calculate the subsequent terms.

Let's assume the first term of the geometric progression is 'a', and the common ratio is 'r'.

Given that the second term is 6, we can write:

a * r = 6 ... Equation (1)

Also, given that the seventh term is 192, we have:

a * r^6 = 192 ... Equation (2)

Now, let's solve these two equations simultaneously.

From Equation (1), we can express 'a' in terms of 'r':

a = 6 / r

Substituting this into Equation (2), we get:

(6 / r) * r^6 = 192
6 * r^5 = 192
r^5 = 192 / 6
r^5 = 32

Taking the fifth root of both sides, we find:

r = ∛(32)
r = 2

Substituting this value back into Equation (1), we can solve for 'a':

a * 2 = 6
a = 6 / 2
a = 3

Now, we can calculate the first three terms.

The first term (a) is 3.
The second term (a * r) is 3 * 2 = 6.
The third term (a * r^2) is 3 * (2^2) = 12.

Therefore, the first three terms of the geometric progression are 3, 6, and 12.

To determine the first three terms of a geometric progression, we need to find the common ratio (r) and the first term (a).

We are given that the second term (a₂) is 6. Let's denote the first term as a₁, the second term as a₂, and the seventh term as a₇.

The formula for finding the nth term of a geometric progression is:
aₙ = a₁ * r^(n-1)

We can use this formula to find a₇:
a₇ = a₁ * r^(7-1)

We are also given that a₇ = 192. Plugging these values into the equation:

192 = a₁ * r^6

Similarly, we can determine the relationship between the second term (a₂) and the first term (a₁):
a₂ = a₁ * r^(2-1)
6 = a₁ * r

We now have two equations:
192 = a₁ * r^6
6 = a₁ * r

To solve these equations, we need to eliminate the variable "a₁". We can divide the second equation by the first equation:

(6/a₁) = (a₁ * r) / (a₁ * r^6)
(6/a₁) = 1 / r^5

Now we can solve for "r":

6 = a₁ * r
(6/a₁) = 1 / r^5

Cross-multiplying both sides:

6 * r^5 = a₁
r^5 = a₁/6

Substituting this value of r into the first equation:

192 = (a₁/6) * (a₁/6)^6
192 = (a₁^7) / 6^7

Simplifying:

192 * 6^7 = a₁^7
a₁^7 = 192 * 6^7

Taking the seventh root of both sides to solve for "a₁":

a₁ = (192 * 6^7)^(1/7)

Now that we have the value of "a₁", we can find "r" using the second equation:

6 = a₁ * r

Substitute the value of "a₁" we just found:

6 = ((192 * 6^7)^(1/7)) * r

Solve for "r":

r = 6 / ((192 * 6^7)^(1/7))

Now that we have the values of "a₁" and "r", we can determine the first three terms by substituting the relevant values into the formula for the nth term of a geometric progression:

a₁ = ((192 * 6^7)^(1/7))
a₂ = a₁ * r
a₃ = a₂ * r

Just substitute the values of "a₁" and "r" into these formulas to find the first three terms of the given geometric progression.