the first, second, and third terms in the geometric progression are k,k-6, 2k-28 respectively. given that all terms of the geometric progression are positive, calculate the value of constant k.
remember that the terms have a constant ratio, so
(2k-28)/(k-6) = (k-6)/k
Now just solve for k. You will get two answers, only one of which makes all the terms positive.
To find the value of constant k in the given geometric progression, we can use the formula for the nth term of a geometric progression:
\(a_n = a_1 \times r^{(n-1)}\)
where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the progression.
Given:
First term (\(a_1\)) = k
Second term (\(a_2\)) = k - 6
Third term (\(a_3\)) = 2k - 28
We can now set up two equations using the formula:
\(k - 6 = k \times r\) ...(Equation 1)
\(2k - 28 = k \times r^2\) ...(Equation 2)
To solve for k, we can substitute Equation 1 into Equation 2:
\(2(k - 6) - 28 = k^2 - 6k\)
Simplifying the equation:
\(2k - 12 - 28 = k^2 - 6k\)
\(2k - 40 = k^2 - 6k\)
Rearranging the equation:
\(k^2 - 6k - 2k + 40 = 0\)
\(k^2 - 8k + 40 = 0\)
Now, we can solve this quadratic equation either by factoring or by using the quadratic formula.
Using the quadratic formula:
\(k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Substituting the values, a = 1, b = -8, and c = 40 into the formula:
\(k = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \times 1 \times 40}}{2 \times 1}\)
\(k = \frac{8 \pm \sqrt{64 - 160}}{2}\)
\(k = \frac{8 \pm \sqrt{-96}}{2}\)
Since all terms in the geometric progression are positive, we have to exclude the negative square root.
\(k = \frac{8 + \sqrt{-96}}{2}\)
We cannot take the square root of a negative number without involving complex numbers. Therefore, there is no real value of k that satisfies the conditions stated in the problem.
To find the value of the constant k, we can use the conditions given in the problem.
We know that the first term of the geometric progression is k.
The second term is k-6.
The third term is 2k-28.
In a geometric progression, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r).
We can use this information to set up equations to find the value of k and r.
From the given information:
Second term / First term = Third term / Second term
(k-6) / k = (2k-28) / (k-6)
Now, let's solve this equation:
Cross-multiply to get rid of the fractions:
(k-6)(k-6) = (2k-28)(k)
Expand both sides of the equation:
k^2 - 12k + 36 = 2k^2 - 28k
Rearrange the equation:
0 = 2k^2 - 28k - k^2 + 12k - 36
Combine like terms:
0 = k^2 - 16k - 36
Factor the quadratic equation:
0 = (k-18)(k+2)
Now, we have two possible solutions:
k - 18 = 0 or k + 2 = 0
Solving each equation for k, we find:
k = 18 or k = -2
Since all the terms of the geometric progression are positive, the value of k must be 18.
Therefore, the value of the constant k is 18.