Answer
a) See table
b) The set of all natural numbers
c) $n=14$
Work Step by Step
We are given the factorial function:
T(n)=1^2+2^2+....+n^2
a) Build a table of the function for n=1,2,...,10.
T(1)=1^2=1
T(2)=1^2+2^2=5
T(3)=1^2+2^2+3^2=14
T(4)=1^2+2^2+...+4^2=30
T(5)=1^2+2^2+...+5^2=55
T(6)=1^2+2^2+...+6^2=91
T(7)=1^2+2^2+...+7^2=140
T(8)=1^2+2^2+...+8^2=204
T(9)=1^2+2^2+...+9^2=285
T(10)=1^2+2^2+...+10^2=385
b) The domain of the function T(n) consists of all the natural numbers (positive integers).
c) Compute T(n) until we reach 1000:
$T(11)=1^2+2^2+...+11^2=506$
$T(12)=1^2+2^2+...+12^2=650$
$T(13)=1^2+2^2+...+13^2=819$
$T(14)=1^2+2^2+...+14^2=1015$
Therefore the least value for which T(n)>1000 is n=14.
