Answer
There will be around 50 sparrows in 2004.
The sparrow will be extinct by 2010.
Work Step by Step
RECALL:
(1) The $n^{th}$ term of a geometric sequence, $a_n$, is represented by the formula:
$a_n=a_1 \cdot r^{n-1}$
where
$a_1$=first term
$r$ = common ratio
$n$ = term number
(2) The next term of a geometric sequence can be found by multiplying the common ratio $r$ to the present term.
$a_n=a_{n-1} \cdot r$
The population of the sparrow can be represented by a geometric sequence with a first term of $800$ and a common ratio of $\frac{1}{2}$ or $0.5$.
The geometric sequence has $a_1=800$ and $r=0.5$.
Note that the year 2004 represents $n=5$ since $n=1$ represents the year $2000$,
Use the formula in (1) above to obtain:
$a_n=a_1 \cdot r^{n-1}
\\a_5=800 \cdot 0.5^{5-1}
\\a_5=800 \cdot 0.5^4
\\a_5=50$
To find the term number when the sparrow will be extinct, just find the succeeding populations to obtain:
$2005: 50(0.5)=25
\\2006: 25(0.5) =12.5 \approx 13
\\2007: 12.5(0.5)=1 \approx 6
\\2008: 6.25(0.5)= 3.125 \approx 3
\\2009: 3.125(0.5) = 1.5625\approx 2
\\2010: 1.5625(0.5) = 0.78125$
This number is already less than $1$ so the sparrow will be extinct by this time.
Therefore, the sparrow will be extinct by the year 2010.
