The sum of infinity and the second term of a geometric progression are 4 and 1 respectively. Find the, 1 common ratio, 2 the first term.
S∞ = a/(1/r) so
a/(1-r) = 4
ar = 1
divide, and you have
r(1-r) = 1/4
r^2 - r + 1/4 = 0
(r - 1/2)^2 = 0
r = 1/2
so a/2 = 1
a = 2
check: 2/(1 - 1/2) = 4
Let's denote the common ratio of the geometric progression as 'r' and the first term as 'a'.
Given that the second term is 1, we can use the formula for the nth term of a geometric progression to find the value of 'a':
n-th term = a * r^(n-1)
For the second term, n = 2:
1 = a * r^(2-1)
1 = a * r
We are also given that the sum of infinity is 4. The sum of an infinite geometric progression can be calculated using the formula:
Sum of infinity = a / (1 - r)
Substituting the given values, we have:
4 = a / (1 - r)
We now have two equations:
1 = a * r
4 = a / (1 - r)
To solve these equations, we can substitute the value of 'a' from the first equation into the second equation:
4 = (1 - r) / r
Now we can solve for 'r':
4r = 1 - r
5r = 1
r = 1/5
Substituting this value of 'r' back into the first equation, we can solve for 'a':
1 = a * (1/5)
a = 5
Therefore, the common ratio of the geometric progression is 1/5 and the first term is 5.
To find the common ratio (r) and the first term (a) of a geometric progression, we need to use the information given.
In a geometric progression, each term is found by multiplying the previous term by a constant called the common ratio (r). Let the first term be "a."
From the information given, we know that the second term is 1. So, the second term can be expressed as:
Second Term = a * r^1
We also know that the sum of infinity, which is the sum of all the terms in the geometric progression, is 4 when it exists. The sum of an infinite geometric progression can be calculated using the formula:
Sum of Infinity = a / (1 - r)
Now, we can use the information to solve for r and a.
1. Finding the common ratio (r):
Substitute the value of the second term (1) into the equation for the second term:
1 = a * r^1
Since the base (r) to the power of 1 is just r, we can rewrite the equation as:
1 = a * r
2. Finding the first term (a):
Substitute the value of the sum of infinity (4) into the equation for the sum of infinity:
4 = a / (1 - r)
To eliminate the denominator, multiply both sides of the equation by (1 - r):
4(1 - r) = a
Distribute 4 to both terms:
4 - 4r = a
Now, we have two equations:
1 = a * r
4 - 4r = a
We can solve this system of equations to find the values of r and a.
To eliminate a, divide the second equation by r:
(4 - 4r) / r = a / r
Simplify the equation:
4/r - 4 = a/r
Substitute the value of a * r (from equation 1) in the second equation:
4/r - 4 = 1/r
Now, we have a single equation involving only r:
4/r - 1/r = 4
Combine the fractions:
(4 - 1)/r = 4
Simplify:
3/r = 4
Cross-multiply:
3 = 4r
Divide both sides by 4 to solve for r:
r = 3/4
Now that we have the value of r, we can substitute it back into equation 1 to find the value of a:
1 = a * (3/4)
Solve for a:
a = 4/3
Therefore, the common ratio (r) is 3/4, and the first term (a) is 4/3.