The 6th term of G.p is 27 and the 12 term is 19,683. Fine the common ratio, the first term and 10th term
These are driving me up the wall.
Please Google:
geometric sequence math is fun
term n = a r^(n-1)
so for example
term 6 = a r^5
To find the common ratio, first, we need to find the first term (a) of the geometric progression (G.P).
Given:
The 6th term (A₆) = 27
The 12th term (A₁₂) = 19,683
The formula for finding the nth term of a geometric progression is:
Aₙ = a * r^(n-1)
Using the formula, we can find the value of 'r' (common ratio):
A₆ = a * r^(6-1) (eq. 1)
27 = a * r^5
A₁₂ = a * r^(12-1) (eq. 2)
19,683 = a * r^11
Divide eq. 2 by eq. 1:
(19,683 / 27) = (a * r^11) / (a * r^5)
Simplifying the equation:
(19,683 / 27) = r^(11-5)
(19,683 / 27) = r^6
Taking the 6th root of both sides:
(r^6)^(1/6) = (19,683 / 27)^(1/6)
r = (19,683 / 27)^(1/6)
Calculating the value of r using a calculator or simplifying it further:
r ≈ 3
Now that we know the value of the common ratio (r), we can find the value of the first term (a) using eq. 1:
27 = a * 3^5
27 = a * 243
a = 27 / 243
a = 1/9
Therefore, the common ratio (r) is 3, the first term (a) is 1/9, and the 10th term can be found using the formula mentioned earlier:
A₁₀ = a * r^(10-1)
A₁₀ = (1/9) * 3^9
Calculating the value of the 10th term:
A₁₀ = 1/9 * 19683
A₁₀ = 2187
Hence, the 10th term is 2187.
To find the common ratio, first let's denote the first term of the geometric progression as 'a' and the common ratio as 'r'.
We know that the 6th term of the G.P is 27, so we can write the equation:
a * r^5 = 27 ---(Equation 1)
Similarly, the 12th term of the G.P is 19,683, so we can write another equation:
a * r^11 = 19,683 ---(Equation 2)
To find the common ratio, we can divide Equation 2 by Equation 1:
(r^11) / (r^5) = 19683 / 27
Simplifying the left side:
r^(11 - 5) = 729
r^6 = 729
Taking the 6th root of both sides:
r = ∛(729) = 3
So, the common ratio (r) is 3.
Now, let's find the first term (a). We can substitute the value of r into Equation 1 to solve for a:
a * (3^5) = 27
3^5 is equal to 3 * 3 * 3 * 3 * 3 = 243, so we can rewrite the equation as:
a * 243 = 27
Dividing both sides by 243:
a = 27 / 243
Simplifying:
a = 1 / 9
So, the first term (a) is 1/9.
Finally, let's find the 10th term of the G.P. We can use the formula for the nth term of a G.P:
Tn = a * r^(n-1)
Substituting the values we found:
T10 = (1/9) * 3^(10-1)
Simplifying:
T10 = (1/9) * 3^9
3^9 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 = 19683, so we can rewrite the equation as:
T10 = (1/9) * 19683
Dividing:
T10 = 2187
Therefore, the 10th term of the G.P is 2187.
To summarize:
- The common ratio (r) is 3.
- The first term (a) is 1/9.
- The 10th term (T10) is 2187.
Nice try, Anonymous.
People who post these basic questions already have the formula. They are just unwilling to do the work. When your blood pressure rises, just scroll on by, and let someone else take 'em on.