4) The sum of an infinite geometric series is 100. The second term of the series is 24. Find two possibilities for a (the
first term) and r (the common ratio).
b. Given,
5 2 4 a a a 159
and
4 1 3 a a a 265
, find the terms of the sequence and give its explicit form
ONLY PART B
a/(1-r) = 100
ar = 24
divide and you have
1/(r(1-r)) = 100/24
r = 2/5 or 3/5
a = 60 or 40
I dont understand
To find the terms of the sequence and give its explicit form, let's solve the equations given:
Equation 1: 5a^2 - 4a + 24 = -159
Equation 2: 4a^2 - a + 3a = 265
Solving Equation 1:
5a^2 - 4a + 24 = -159
Rearranging the equation:
5a^2 - 4a + 183 = 0
Using the quadratic formula:
a = (-(-4) ± √((-4)^2 - 4*(5)*(183))) / (2*(5))
a = (4 ± √(16 - 3660)) / 10
a = (4 ± √(-3644)) / 10
Since the discriminant is negative, there are no real solutions for a in Equation 1.
Solving Equation 2:
4a^2 - a + 3a = 265
Simplifying the equation:
4a^2 + 2a - 265 = 0
Using the quadratic formula:
a = (-2 ± √(2^2 - 4*(4)*(-265))) / (2*(4))
a = (-2 ± √(4 + 4240)) / 8
a = (-2 ± √(4244)) / 8
a = (-2 ± 65.15) / 8
Taking both the positive and negative roots for a, we get two possibilities:
a₁ = (-2 + 65.15) / 8 = 63.15 / 8 ≈ 7.89
a₂ = (-2 - 65.15) / 8 = -67.15 / 8 ≈ -8.39
Therefore, the terms of the sequence and its explicit form for both possibilities of a are as follows:
For a = 7.89:
Term 1: a₁ = 7.89
Term 2: a₁ * r = 7.89 * r
Term 3: (a₁ * r) * r = 7.89 * r^2
And so on...
For a = -8.39:
Term 1: a₂ = -8.39
Term 2: a₂ * r = -8.39 * r
Term 3: (a₂ * r) * r = -8.39 * r^2
And so on...
Note that the common ratio (r) is still unknown and needs to be determined separately.
To find the terms of the sequence and give its explicit form, we can start by solving the given equations:
Equation 1: 5a^2 - 4a - 159 = 0
Equation 2: 4a^3 - a^2 - 265 = 0
Let's solve Equation 1 first:
1. We can factor the equation by finding two numbers that multiply to -159 and add up to -4.
The factors of 159 are: 1, 3, 53, and 159.
Out of these factors, we can see that 53 and -3 add up to -4.
So, the factored form of Equation 1 is: (a - 53)(a + 3) = 0
Now, let's solve Equation 2:
2. Since Equation 2 is a cubic equation, it might not have a simple factored form. We can solve it numerically using a calculator or by using iterative methods like the Newton-Raphson method.
After solving Equation 2, we find three possible solutions:
a ≈ 4.144, a ≈ -1.626, and a ≈ 2.483
Now that we have the possible values for 'a', let's find the sequence terms.
For the explicit form of the sequence, we'll use the formula for the nth term of an infinite geometric series:
tn = a * r^(n - 1)
Where 'a' is the first term and 'r' is the common ratio.
To find the common ratio 'r', we can use the given information: the sum of the infinite geometric series is 100, and the second term is 24.
The sum of an infinite geometric series is given by the formula:
S = a / (1 - r)
Given that S = 100 and the second term is 24, we can set up the following equation:
100 = a / (1 - r)
100 = a / (1 - (a * r))
We can solve this equation for r using the value of 'a' we found from Equation 1.
Once we have the value of 'r', we can find the terms of the sequence by substituting the values of 'a' and 'r' into the explicit form formula.
So, the explicit form of the sequence is tn = a * r^(n - 1), where 'a' and 'r' are the values we found.