The sum of the first four terms of a G.P.is 30 and that of the last four terms is 960. If the first and the last term of the G.P. are 2 and 512 respectively, find the the common ratio.
first term is 2
so a = 2
last term is 512
ar^(n-1) = 512
2r^(n-1) = 512
r^(n-1) = 256
a + ar + ar^2 + ar^3 = 30 , but a=2
2(1 + r + r^2 + r^3) = 30
(1 + r + r^2 + r^3) = 15 (equation #1)
ar^(n-1) + ar^(n-2)+ar^(n-3)+ar^(n-4) = 960 , remember a=2
r^(n-1) + r^(n-2)+r^(n-3)+r^(n-4) = 480
r^(n-4)(r^3 + r^2 + r + 1) = 480 (equation #2)
divide #2 by #1 ----> r^(n-4) = 480/15 = 32
now r^(n-1) / r^(n-4) = 256/32 = 8
r^3 = 8
r = 2
The sum of three number in G.P.is 14 and their product is -216. find the number.
To find the common ratio of a geometric progression (G.P.), we can use the formula:
\[S_n = \dfrac{a(r^n - 1)}{r - 1}\]
where \(S_n\) is the sum of the first \(n\) terms, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
Given that the first term \(a\) is 2 and the last term is 512, we can determine the number of terms (\(n\)) by using the formula:
\[a_n = ar^{n-1}\]
Substituting the values, we have:
\[512 = 2r^{n-1}\]
To simplify the problem, we will take the logarithm of both sides:
\[\log(512) = \log(2r^{n-1})\]
Applying the logarithmic property \(\log(x^y) = y \log(x)\), we have:
\[9 = (n-1) \log(r)\]
Simplifying further, we can write:
\[\log(r) = \dfrac{9}{n-1}\]
Now, let's solve for the common ratio.
To find the sum of the first four terms of the G.P., we can use the formula:
\[S_4 = \dfrac{a(r^4 - 1)}{r - 1}\]
Given that \(S_4 = 30\) and \(a = 2\), we can substitute these values into the equation:
\[30 = \dfrac{2(r^4 - 1)}{r - 1}\]
Simplifying further, we have:
\[15(r - 1) = r^4 - 1\]
\[15r - 15 = r^4 - 1\]
The next step is to find the sum of the last four terms of the G.P., which is given as 960.
Using the formula for the sum of the last four terms:
\[S_{\text{last 4}} = \dfrac{a(r^n - 1)}{r - 1}\]
Given that \(S_{\text{last 4}} = 960\), \(a = 512\), and \(n \geq 4\), we can substitute these values into the equation:
\[960 = \dfrac{512(r^n - 1)}{r - 1}\]
Simplifying further, we have:
\[480(r - 1) = r^n - 1\]
\[480r - 480 = r^n - 1\]
At this point, we have two equations with two unknowns:
Equation 1: \(15r - 15 = r^4 - 1\)
Equation 2: \(480r - 480 = r^n - 1\)
By solving these two equations simultaneously, we can find the values of \(r\) and \(n\).