Answer
$2801$
Work Step by Step
Counting
1. the man, $a_{1}=1,$
2. the wives, $a_{2}=7,$
3. the sacks, $a_{3}=7a_{2}=7^{2}=49,$
4. the cats, $ a_{4}=7a_{3}=7^{3},$
5. the kits, $a_{5}=7a_{4}=7^{4},$
(common ratio =7),
we have a geometric sequence with a=1, r=7,
$a_{n}=1\cdot 7^{n-1}.$
The total is
$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=\displaystyle \sum_{k=1}^{5}ar^{k-1},$
the partial sum of a geometric sequence,
$S_{n}=a\displaystyle \cdot\frac{1-r^{n}}{1-r}$, where n=5, a=1, r=7.
$S_{5}=1\displaystyle \cdot\frac{1-7^{5}}{1-7}=\frac{16,806}{6}=2801$.
