In a geometric progression (GP), the terms can be expressed as follows:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)
Given:
- The second term \( ar = 16 \)
- The fifth term \( ar^4 = 2 \)
We can use these two equations to find the values of \( a \) and \( r \).
From the first equation, we can express \( a \) in terms of \( r \): \[ a = \frac{16}{r} \]
Now, substitute \( a \) into the second equation \( ar^4 = 2 \): \[ \left(\frac{16}{r}\right) r^4 = 2 \] \[ 16r^3 = 2 \] \[ r^3 = \frac{2}{16} = \frac{1}{8} \] \[ r = \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2} \]
Now that we have \( r \), we can find \( a \): \[ a = \frac{16}{r} = \frac{16}{\frac{1}{2}} = 16 \times 2 = 32 \]
Thus, the first term and the common ratio of the geometric progression are:
- First term \( a = 32 \)
- Common ratio \( r = \frac{1}{2} \)
So, the answer is:
- First term: \( 32 \)
- Common ratio: \( \frac{1}{2} \)
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