THE 8th term of a g.p is -7/32 calculate the common ratio if the terms is 28
what do you mean "the terms is 28" ??
However, -7/32 = 28 * (-1/2)^7
Math
To find the common ratio (r) of a geometric progression (g.p.), you need two terms of the g.p. Using the 8th term (-7/32) and the 28th term, we can calculate the common ratio.
Let's denote the first term of the g.p. as 'a' and the common ratio as 'r'.
The formula for the nth term of a g.p. is given by:
\(T_n = a \cdot r^{(n-1)}\)
Given that the 8th term (T8) is -7/32 and the 28th term (T28) is 28, we can set up two equations:
For the 8th term:
\(T_8 = a \cdot r^{(8-1)} = -\frac{7}{32}\) ..........(1)
For the 28th term:
\(T_{28} = a \cdot r^{(28-1)} = 28\) ..........(2)
Dividing equation (2) by equation (1), we get:
\(\frac{T_{28}}{T_8} = \frac{28}{-\frac{7}{32}}\)
Simplifying this expression, we have:
\(r^{28-8} = -4 \)
\(r^{20} = -4\)
Now, to find the common ratio, we can take the 20th root of both sides:
\(r = \sqrt[20]{-4} \)
Since we are dealing with real numbers, the 20th root of -4 is not a real number. Therefore, we do not have a valid common ratio for this g.p.
To find the common ratio of a geometric progression (g.p), we can use the formula:
\[a_n = a_1 \cdot r^{(n-1)}\]
where \(a_n\) represents the \(n\)th term of the g.p, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term we want to find.
In this case, we are given that the 8th term (\(a_8\)) is \(-\frac{7}{32}\), and the value of this term is 28. We can use this information to solve for the common ratio.
Setting this up, we have:
\[-\frac{7}{32} = a_1 \cdot r^{(8-1)}\]
Simplifying, we get:
\[-\frac{7}{32} = a_1 \cdot r^7\]
Since we don't have the value of the first term (\(a_1\)), we can express it in terms of the value of the 7th term (\(a_7\)) using the same formula. So, we have:
\[28 = a_1 \cdot r^{(7-1)} = a_1 \cdot r^6\]
Now we have a system of equations:
\[-\frac{7}{32} = a_1 \cdot r^7\]
\[28 = a_1 \cdot r^6\]
To solve this system, we can divide the second equation by the first equation:
\[\frac{28}{-\frac{7}{32}} = \frac{a_1 \cdot r^6}{a_1 \cdot r^7}\]
\[-128 = \frac{1}{r}\]
\[\frac{1}{r} = -128\]
Taking the reciprocal of both sides, we get:
\[r = -\frac{1}{128}\]
Therefore, the common ratio of the geometric progression is \(-\frac{1}{128}\).