Question 549647
The trouble here is that if your 11th term is t(11)=72, your "initial value t(0) is really your 0th term. I call the 11th term {{{a[11]}}}, the first term {{{a[1]}}}, and I do not acknowledge the existence of an initial value, or term number 0. I'll have to translate my thinking to your way of naming things.
In your way of naming terms:
The common difference is {{{d}}}.
There is an initial value {{{t(0)}}}, but
the first term is {{{t(1)=t(0)+d}}}.
The second term is {{{t(2)=t(1)+d}=t(0)+2d}}.
The third term is {{{t(3)=t(2)+d=t(0)+3d}}}.
The n-th term is {{{t(n)=t(0)+nd}}}.
The 11th term is {{{t(11)=t(0)+11d=72}}}.
The 27th term is {{{t(27)=t(0)+27d=96}}}.
So {{{t(27)-t(11)=t(0)+27d-(t(0)+11d)=t(0)+27d-t(0)-11d=16d}}}.
But {{{t(27)-t(11)=96-72=24}}},
so {{{16d=24}}} --> {{{d=24/16=3/2}}}
The common difference is {{{3/2=1&1/2}}}, and
{{{t(11)=t(0)+11d=t(0)+11*(3/2)=72}}}.
So, {{{t(0)+11*(3/2)=72}}} --> {{{t(0)=72-11*(3/2)=72-33/2=144/2-33/2=111/2=55&1/2}}}
(And the first term is {{{t(1)=t(0)+d=55&1/2+1&1/2=57}}}).
JUST IN CASE someone whats to read this in my language
In my way of naming terms:
The common difference is {{{d}}}.
The first term is {{{a[1]}}}.
The second term is {{{a[2]=a[1]+d}}}.
The third term is {{{a[3]=a[2]+d=a[1]+2d}}}.
The n-th term is {{{a[n]=a[1]+(n-1)d}}}.
The 11th term is {{{a[11]=a[1]+10d=72}}}.
The 27th term is {{{a[27]=a[1]+26d=96}}}.
So {{{a[27]-a[11]=a[1]+26d-(a[1]+10d)=a[1]+26d-a[1]-10d=16d}}}.
But {{{a[27]-a[11]=96-72=24}}},
so {{{16d=24}}} --> {{{d=24/16=3/2}}}
The common difference is {{{3/2}}}
and {{{a[11]=a[1]+10d=a[1]+10*(3/2)=a[1]+15=72}}} --> {{{a[1]=72-15=57}}}