find sum of the first 10terms of geometric series with a2=1/2 and r=2
To find the sum of the first 10 terms of a geometric series, we can use the formula:
\[S_n = \frac{a_1(1 - r^n)}{1 - r}\]
Where:
- \(S_n\) is the sum of the first \(n\) terms of the geometric series.
- \(a_1\) is the first term of the series.
- \(r\) is the common ratio between the terms.
- \(n\) is the number of terms we want to sum.
In this case, we are given \(a_2 = \frac{1}{2}\) and \(r = 2\). However, we need to find \(a_1\) in order to use the formula. We can do this by finding the common ratio between \(a_1\) and \(a_2\).
We know that:
\[a_2 = a_1 \cdot r\]
Substituting in the given values:
\[\frac{1}{2} = a_1 \cdot 2\]
Solving for \(a_1\):
\[a_1 = \frac{1}{2 \cdot 2} = \frac{1}{4}\]
Now we can substitute the values of \(a_1\), \(r\), and \(n\) into the formula to find the sum of the first 10 terms:
\[S_{10} = \frac{\frac{1}{4}(1 - 2^{10})}{1 - 2}\]
\[S_{10} = \frac{\frac{1}{4}(1 - 1024)}{-1}\]
\[S_{10} = \frac{\frac{1}{4}(-1023)}{-1}\]
\[S_{10} = \frac{-1023}{4}\]
Therefore, the sum of the first 10 terms of the geometric series is \(\frac{-1023}{4}\) or -255.75.