Answer
a) See table
b) The set of all natural numbers
c) $n=45$
Work Step by Step
We are given the factorial function:
$S(n)=1+2+....+n$
a) Build a table of the function for n=1,2,...,10.
$f(1)=1$
$f(2)=1+2=3$
$f(3)=1+2+3=6$
$f(4)=1+2+3+4=10$
$f(5)=1+2+3+4+5=15$
$f(6)=1+2+3+4+5+6=21$
$f(7)=1+2+3+4+5+6+7=28$
$f(8)=1+2+3+4+5+6+7+8=36$
$f(9)=1+2+3+4+5+6+7+8+9=45$
$f(10)=1+2+3+4+5+6+7+8+9+10=55$
b) The domain of the function $S(n)$ consists of all the natural numbers (positive integers).
c) Compute $S(n)$ until we reach 1000:
$S(30)=1+2+...+30=465$
$S(40)=1+2+...+40=820$
$S(41)=1+2+...+41=861$
$S(42)=1+2+...+42=903$
$S(43)=1+2+...+43=946$
$S(44)=1+2+...+44=990$
$S(45)=1+2+...+45=1035$
Therefore the least value for which S(n)>1000 is n=45.
