The second term of a GP is 1/2 and the sum ti infinity of the series is 4. Find the first term and and the common ratio of the series.
ar = 1/2
a/(1-r) = 4
a = 2-√2
r = 1/4 (2+√2)
or
a = 2+√2
r = 14 (2-√2)
Well, isn't this a puzzling series! It seems like our geometric progression might have a bit of an identity crisis.
Let's call the first term of our GP "a," and the common ratio "r." According to the given information, the second term is 1/2, which means our GP looks something like this: a, 1/2, ...
Now, the sum to infinity of a geometric progression can be found using the formula S = a / (1 - r), where S is the sum and r is the common ratio. In our case, the sum to infinity is 4, so we have:
4 = a / (1 - r)
Now, let's use a little algebraic magic to solve this mystery. Multiplying both sides by (1 - r), we get:
4(1 - r) = a
Expanding, we have:
4 - 4r = a
So, we've figured out that the first term "a" is equal to 4 - 4r. But what about the second term, which we know is 1/2? Let's plug that in and solve for "r":
1/2 = (4 - 4r) * r
We can simplify this equation to:
1/2 = 4r - 4r^2
Now, let's put on our detective hats and solve this quadratic equation:
4r^2 - 4r + 1/2 = 0
Solving this equation, we find two possible solutions for "r." But being a clown, I won't bore you with all the gory details. I'll just reveal the answers:
The possible values for "r" are approximately 1/4 and 1. Yes, you heard it right, there are two solutions!
Now, with these values of "r," we can go back and find the first term "a" using our earlier equation:
a = 4 - 4r
Substituting the values of "r," we get:
a = 4 - 4(1/4) = 3
And
a = 4 - 4(1) = 0
So, we have two possible answers! The first term could be either 3 or 0, depending on which common ratio we decide to use.
There you have it! We've discovered the mysterious first term and possible common ratios of our perplexing geometric progression.
To find the first term and the common ratio of the geometric progression (GP), we will use the given information.
Let's denote the first term as 'a' and the common ratio as 'r'.
Given:
Second term, a₂ = 1/2
Sum to infinity of the series, S = 4
The formula to calculate the sum to infinity of a geometric series is:
S = a / (1 - r)
We are given that S = 4, so we have:
4 = a / (1 - r)
Also, we are given that the second term, a₂ = 1/2, so we can write:
a₂ = a * r₁
1/2 = a * r
Now we have two equations:
1. 4 = a / (1 - r)
2. 1/2 = a * r
To solve these equations simultaneously, we can rearrange the second equation to express 'a' in terms of 'r':
a = (1/2) / r
Substituting this value of 'a' into the first equation, we get:
4 = ((1/2) / r) / (1 - r)
Multiplying through by r and (1 - r), we obtain:
4r - 4r² = 1/2
Multiplying through by 2 to eliminate the fraction, we have:
8r - 8r² = 1
Rearranging the equation to obtain a quadratic equation in standard form:
8r² - 8r + 1 = 0
Now we can solve this quadratic equation to find the values of 'r'.
Using the quadratic formula:
r = (-b ± √(b² - 4ac)) / (2a)
For the equation 8r² - 8r + 1 = 0, the values of 'a', 'b', and 'c' are:
a = 8
b = -8
c = 1
Substituting these values into the quadratic formula, we have:
r = (-(-8) ± √((-8)² - 4(8)(1))) / (2(8))
r = (8 ± √(64 - 32)) / 16
r = (8 ± √32) / 16
r = (8 ± 4√2) / 16
r = (1 ± √2) / 2
So, the possible values of 'r' are (1 + √2) / 2 and (1 - √2) / 2.
Now, we can substitute these values back into the equation a = (1/2) / r to find the first term 'a'.
For r = (1 + √2) / 2:
a = (1/2) / ((1 + √2) / 2)
a = 1 / (1 + √2)
For r = (1 - √2) / 2:
a = (1/2) / ((1 - √2) / 2)
a = 1 / (1 - √2)
Therefore, the first term and the common ratio of the geometric progression are:
If r = (1 + √2) / 2:
First term, a = 1 / (1 + √2)
Common ratio, r = (1 + √2) / 2
If r = (1 - √2) / 2:
First term, a = 1 / (1 - √2)
Common ratio, r = (1 - √2) / 2
To find the first term and common ratio of the geometric progression (GP), we can use the given information:
Given:
- The second term of the GP is 1/2.
- The sum to infinity (also known as the sum of an infinite geometric series) is 4.
Let's denote the first term of the GP as 'a' and the common ratio as 'r'.
The formula for the sum to infinity of a GP is:
Sum = a / (1 - r)
We are told that the sum to infinity is 4, so we can write the equation:
4 = a / (1 - r) ----(1)
We are also given that the second term of the GP is 1/2. This allows us to create another equation involving 'a' and 'r'.
The second term, a₂, is given by:
a₂ = a * r ----(2)
Substituting the given value for a₂ (1/2) into equation (2), we get:
1/2 = a * r ----(3)
Now we have a system of equations (1) and (3) that we can solve simultaneously to find the values of 'a' and 'r'.
Let's solve the system of equations:
From equation (3), we can solve for 'a' in terms of 'r':
a = (1/2) / r
Substituting this value of 'a' into equation (1), we get:
4 = ((1/2) / r) / (1 - r)
To simplify, multiply both sides of the equation by r(1 - r):
4r(1 - r) = 1/2
Expanding and rearranging the equation:
4r - 4r² = 1/2
Multiplying through by 2 to eliminate the fraction:
8r - 8r² = 1
Rearranging the equation and setting it equal to zero:
8r² - 8r + 1 = 0
This is now a quadratic equation in 'r'. To solve for 'r', we can use the quadratic formula:
r = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = 8, b = -8, and c = 1. Substituting these values into the quadratic formula, we get:
r = (-(-8) ± √((-8)² - 4(8)(1))) / (2(8))
r = (8 ± √(64 - 32)) / 16
r = (8 ± √32) / 16
Simplifying further:
r = (8 ± 4√2) / 16
r = (1 ± 0.5√2) / 2
Now we have two possible values for 'r'.
Substituting each value of 'r' back into equation (3), we can solve for the corresponding value of 'a'.
Case 1: r = (1 + 0.5√2) / 2
Substituting into equation (3):
1/2 = a * [(1 + 0.5√2) / 2]
Simplifying and rearranging:
1 = a * (1 + 0.5√2)
a = 1 / (1 + 0.5√2)
Case 2: r = (1 - 0.5√2) / 2
Substituting into equation (3):
1/2 = a * [(1 - 0.5√2) / 2]
Simplifying and rearranging:
1 = a * (1 - 0.5√2)
a = 1 / (1 - 0.5√2)
Therefore, the first term and common ratios of the series are as follows:
- Case 1: First term (a) = 1 / (1 + 0.5√2), Common ratio (r) = (1 + 0.5√2) / 2
- Case 2: First term (a) = 1 / (1 - 0.5√2), Common ratio (r) = (1 - 0.5√2) / 2