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Consider the sequence $\frac{1}{2}, 4, \frac{1}{4}, 7, \frac{1}{8}, 10, \cdots$. Calculate the sum of the first 50 yterms of that sequence
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Combination of arithmetic and geometric progressions

so $\frac{1}{2},\frac{1}{4}, \frac{1}{8},\ldots$ follows a geometric series with common ratio $r=\frac{1}{2}$

and $4,7,10,\ldots$ with common difference $d=3$

The sum of the first 50 terms will be 25 terms of the geometric series and 25 terms of the arithmetic series

Now, sum of geometric series is given by the formula

$S_n = \frac{a(1-r^n)}{1-r}$

so the sum of the first 25 terms will be

$S_{10} = \frac{\frac{1}{2}(1-{\frac{1}{2}}^{25})}{1-{\frac{1}{2}}}$

$=1$

then the sum of the arithmetic series is given by the formula

$S_n=\frac{n}{2}[2a+(n-1)d]$

so the sum of the first 25 terms will be

$S_{25}=\frac{25}{2}[2(4)+(25-1)\times{3}]$

$=1 000$

Therefore, thesum of the first 50 terms of the series will be

$1 000 + 1 = 1 001$
by Wooden (162 points)

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