a geometric progression has a third term of 20 and a sum to infinity which is three times the first term. find the first term
Is this a follow up from your previous question?
term(3) = ar^2 = 20
a = 20/r^2
a/(1-r) = 3a
1/(1-r) = 3
same as in your previous post, r = 2/3
then a = 20/(4/9(
= 20(9/4) = 45
test:
your sequence is 45 , 30 , 20 , 40/3 , 80/9 , ...
is the third term 20 ? , YES
sum of all terms = 20/(1-2/3)
= 20/(1/3)
= 60 , which is 3 times the first term.
All is good!
To find the first term of the geometric progression, we need to use the given information:
1. The third term of the geometric progression is 20.
2. The sum to infinity of the geometric progression is three times the first term.
Let's solve this step by step:
Step 1: Determine the common ratio (r) of the geometric progression.
The third term of the geometric progression can be expressed as:
a * r^2 = 20, where 'a' is the first term and 'r' is the common ratio.
Step 2: Express the sum to infinity of the geometric progression.
The formula for the sum to infinity of a geometric progression is:
S = a / (1 - r), where 'S' is the sum to infinity, 'a' is the first term, and 'r' is the common ratio.
Given that the sum to infinity is three times the first term:
S = 3a.
Step 3: Substitute the given information.
Since S = 3a, we can rewrite the formula for the sum to infinity equation as:
3a = a / (1 - r).
Step 4: Substitute the third term information.
Since a * r^2 = 20, we can rewrite the formula for the common ratio equation as:
r^2 = 20 / a.
Step 5: Substitute the expression for the common ratio (r^2) in the sum to infinity equation.
Now, substitute r^2 = 20 / a into the equation 3a = a / (1 - r):
3a = a / (1 - 20 / a).
Simplify the equation by cross-multiplication:
3a = a(a - 20) / a.
3a = a - 20.
Step 6: Solve for 'a', the first term.
Rearrange the equation:
3a - a = 20.
2a = 20.
a = 10.
Therefore, the first term of the geometric progression is 10.
To find the first term of the geometric progression, we can use the following steps:
Step 1: Understand the problem:
We are given that the third term of the geometric progression is 20, and the sum to infinity is three times the first term. We need to find the first term of the progression.
Step 2: Set up variables:
Let's assume that the first term of the geometric progression is "a", and the common ratio is "r".
Step 3: Use the given information:
We know that the third term of the geometric progression is given by:
Third term (T3) = a * r^2 = 20 ----- (1)
We are also given that the sum to infinity (S∞) is three times the first term:
S∞ = a / (1 - r) = 3a ----- (2)
Step 4: Solve the equations:
From equation (1), we can rewrite it as:
r^2 = 20 / a
Substituting this in equation (2), we get:
a / (1 - (20 / a)) = 3a
Simplifying the equation further:
a / (1 - (20 / a)) = 3a
a / ((a - 20) / a) = 3a
a^2 / (a - 20) = 3a
Cross-multiplying:
a^2 = 3a * (a - 20)
a^2 = 3a^2 - 60a
Rearranging the equation:
2a^2 - 60a = 0
Taking out common factors, we get:
2a(a - 30) = 0
Therefore, a = 0 or a - 30 = 0
Since the first term of a geometric progression cannot be zero, we can disregard a = 0.
Therefore, the first term of the geometric progression (a) is:
a = 30
So, the first term of the geometric progression is 30.
Why did the geometric progression go to therapy?
Because it couldn't stop growing exponentially!
Now, let's get down to business. The formula for the sum of an infinite geometric progression is:
S = a / (1 - r),
Where S is the sum, a is the first term, and r is the common ratio. In this case, we know that the sum is three times the first term, so:
S = 3a.
We also know that the third term is 20. In a geometric progression, the third term can be expressed as:
a * r^2 = 20.
Now, let's solve for the first term (a):
a / (1 - r) = 3a.
Cross-multiplying:
1 - r = 3.
Simplifying:
r = 1 - 3 = -2.
Now, let's substitute the value of r back into the equation for the third term:
a * (-2)^2 = 20.
Simplifying:
4a = 20.
Dividing both sides by 4:
a = 5.
Ta-da! The first term of the geometric progression is 5.