The third term of a geometric progression of positive terms is 18 and the fifth term is 162
Find the common ration and the first term?
Write an expression for the Nth term of the progression?
Find the sum of the first 8 terms of the progression??
To find the common ratio and the first term of the geometric progression, we can use the given information that the third term is 18 and the fifth term is 162.
Step 1: Find the common ratio (r):
The ratio between consecutive terms in a geometric progression is constant. We can find the common ratio by dividing any term by the previous term. Let's divide the fifth term by the third term:
162 / 18 = 9
So, the common ratio (r) is 9.
Step 2: Find the first term (a):
To find the first term, we can use the formula for the nth term of a geometric progression. Let's substitute the values into the formula:
a * r^(n-1) = nth term
Using the fifth term:
a * (9)^(5-1) = 162
a * 9^4 = 162
a * 6561 = 162
a = 162 / 6561
a = 0.0247
Therefore, the common ratio (r) is 9 and the first term (a) is approximately 0.0247.
Step 3: Write an expression for the nth term:
The nth term formula for a geometric progression is:
Tn = a * r^(n-1)
So, the expression for the nth term of this geometric progression is:
Tn = 0.0247 * (9)^(n-1)
Step 4: Find the sum of the first 8 terms:
To find the sum of the first 8 terms of the geometric progression, we can use the formula for the sum of a finite geometric series. The formula is:
Sn = a * (r^n - 1) / (r - 1)
Substituting the values into the formula:
Sn = 0.0247 * (9^8 - 1) / (9 - 1)
Sn = 0.0247 * (43046721 - 1) / 8
Sn = 0.0247 * 43046720 / 8
Sn = 1066701.6
Therefore, the sum of the first 8 terms of the geometric progression is approximately 1066701.6.
To find the common ratio and the first term of a geometric progression, we can use the given information about the third and fifth terms.
Let's start by finding the common ratio (r).
We know that the third term is 18, so let's call it a₃.
We also know that the fifth term is 162, so let's call it a₅.
The formula for the nth term of a geometric progression is given by:
aₙ = a₁ * r^(n-1)
Using this formula, we can write two equations based on the given information:
a₃ = a₁ * r^2 --- equation 1
a₅ = a₁ * r^4 --- equation 2
Now, divide equation 2 by equation 1 to eliminate a₁ and solve for r:
(a₅ / a₃) = (a₁ * r^4) / (a₁ * r^2)
162 / 18 = r^2
9 = r^2
Taking the square root of both sides, we find:
r = ±3
Since we are dealing with a geometric progression of positive terms, the common ratio (r) cannot be negative. Therefore, the common ratio is 3.
Now, to find the first term (a₁), we can substitute the value of r into either equation 1 or equation 2. Let's use equation 1:
18 = a₁ * 3^2
18 = 9a₁
Divide both sides by 9:
a₁ = 2
So, the common ratio is 3 and the first term is 2.
To find the expression for the nth term of the progression, we can substitute the values of a₁ and r into the formula we mentioned earlier:
aₙ = a₁ * r^(n-1)
aₙ = 2 * 3^(n-1)
Now, to find the sum of the first 8 terms of the progression, we'll use the formula for the sum of a geometric series:
Sₙ = a₁ * (1 - r^n) / (1 - r)
Substituting the values of a₁, r, and n:
S₈ = 2 * (1 - 3^8) / (1 - 3)
Evaluate this expression to find the sum of the first 8 terms of the progression.
r^2 = 162/18 = 9
Now you can find the 1st term, and then
S8 = a(r^8-1)/(r-1)