Which term of the G.P. :
`-10,(5)/(sqrt(3)),-(5)/(6), . . . . . .. . . . " is "-(5)/(72)` ?

Answer

To find which term of the geometric progression (G.P.) is equal to \(-\frac{5}{72}\), we will follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \(a\) of the G.P. is given as:
\[ a = -10 \]

To find the common ratio \(r\), we take the second term and divide it by the first term:
\[ r = \frac{\text{Second term}}{\text{First term}} = \frac{\frac{5}{\sqrt{3}}}{-10} \]
Calculating this gives:
\[ r = \frac{5}{\sqrt{3}} \times \frac{-1}{10} = -\frac{1}{2\sqrt{3}} \]

### Step 2: Use the formula for the nth term of a G.P.
The nth term \(T_n\) of a G.P. can be expressed as:
\[ T_n = a \cdot r^{n-1} \]
We know that \(T_n = -\frac{5}{72}\). Therefore, we can set up the equation:
\[ -\frac{5}{72} = -10 \cdot \left(-\frac{1}{2\sqrt{3}}\right)^{n-1} \]

### Step 3: Simplify the equation
First, we can eliminate the negative signs:
\[ \frac{5}{72} = 10 \cdot \left(-\frac{1}{2\sqrt{3}}\right)^{n-1} \]
Now, divide both sides by 10:
\[ \frac{5}{720} = \left(-\frac{1}{2\sqrt{3}}\right)^{n-1} \]

### Step 4: Simplify \(\frac{5}{720}\)
\[ \frac{5}{720} = \frac{1}{144} \]
So we have:
\[ \frac{1}{144} = \left(-\frac{1}{2\sqrt{3}}\right)^{n-1} \]

### Step 5: Express \(\frac{1}{144}\) in terms of powers
We can express \(\frac{1}{144}\) as:
\[ \frac{1}{144} = \left(\frac{1}{12}\right)^2 \]
Now we need to express \(-\frac{1}{2\sqrt{3}}\) in a similar way. We can find its magnitude:
\[ \left(-\frac{1}{2\sqrt{3}}\right)^{n-1} = \frac{1}{(2\sqrt{3})^{n-1}} \]

### Step 6: Set the powers equal
Now we equate the magnitudes:
\[ (2\sqrt{3})^{n-1} = 144 \]
Since \(144 = 12^2\), we can express \(12\) in terms of \(2\) and \(\sqrt{3}\):
\[ 12 = 2^2 \cdot 3^{1/2} \]
Thus,
\[ (2\sqrt{3})^{n-1} = (2^2 \cdot 3^{1/2})^2 = 2^4 \cdot 3^2 \]
This gives us:
\[ 2^{n-1} \cdot 3^{(n-1)/2} = 2^4 \cdot 3^2 \]

### Step 7: Equate the powers
From the powers of \(2\):
\[ n-1 = 4 \implies n = 5 \]
From the powers of \(3\):
\[ \frac{n-1}{2} = 2 \implies n-1 = 4 \implies n = 5 \]

### Conclusion
Thus, the term \(-\frac{5}{72}\) is the 5th term of the G.P.