+1622 Find the sum of \(1 - \frac{1}{2} + \frac{1}{5} - \frac{1}{10} + \frac{1}{25} - \frac{1}{50} + \frac{1}{125} - \frac{1}{250} + \cdots + \frac{1}{5^{10}} - \frac{1}{2 \cdot 5^{10}}\).
The answer should be in the form of a decimal rounded to the nearest HUNDREDTH.
I think this problem is looking for a few equations related to the sequence, but not sure which ones. Thanks :/
+118608
\(1 - \frac{1}{2} + \frac{1}{5} - \frac{1}{10} + \frac{1}{25} - \frac{1}{50} + \frac{1}{125} - \frac{1}{250} + \cdots + \frac{1}{5^{10}} - \frac{1}{2 \cdot 5^{10}}\\ =1 + \frac{1}{5} + \frac{1}{25} +\frac{1}{125} + \cdots + \frac{1}{5^{10}} \\ - \frac{1}{2} - \frac{1}{10} - \frac{1}{50} - \frac{1}{250} + \cdots - \frac{1}{2 \cdot 5^{10}}\)
the first line is the sum of a GP a=1, r=1/5 11 terms
the second is the sum of a GP a=-1/2, r=1/5 11 terms (that dot is a times sign not a decimal point)
Work them out and add them together.
+1622 Thanks for the initial step Melody.
So for the first line I got 1.2499999744.
Then subtracting from the first line with the second line I got very close guess. Instead of subtracting everything else. I got to 0.6250015744, and I knew rounding to the nearest hundredth would get me 0.63 or 0.62.
With your strategy, I got 0.62 as the nearest hundredth.
