Question 732668
{{{a}}}= first term
{{{d}}}= common difference ({{{d<>0}}})
 
{{{a+d}}}= second term
{{{a+(8-1)d=a+7d}}}= 8th term
second term + 8th term ={{{a+d+a+7d=2a+8d=2(a+4d)=20}}}
{{{2(a+4d)=20}}} --> {{{a+4d=10}}} <--> {{{a=10-4d}}}
 
fifth term ={{{a+(5-1)d=a+4d=10}}}
tenth term ={{{a+(10-1)d=a+9d}}}
The first, fifth and tenth terms are in geometric progression, so the common ratio of that geometric progression is
{{{10/a=(a+9d)/10}}} --> {{{10*10=a(a+9d)}}} --> {{{100=a(a+9d)}}}
Substituting  {{{a=10-4d}}} we get
{{{100=(10-4d)(10-4d+9d)}}} --> {{{100=(10-4d)(10+5d)}}} --> {{{100=100+10d-20d^2}}} --> {{{0=10d-20d^2}}} --> {{{0=10d(1-2d)}}}
Since {{{d<>0}}} it must be {{{1-2d=0}}} --> {{2d=1}}} --> {{{highlight(d=1/2)}}}
and {{{a=10-4(1/2)}} --> {{{a=10-2}}} --> {{{highlight(a=8)}}}