Question 743400
{{{a[1]}}}= first term
{{{d}}}= common difference
{{{S}}}= sum of all 10 terms
 
The sum of all terms of the arithmetic progression except for the first term is 99 translates into
{{{S-a[1]=99}}}
The sum of all terms of the arithmetic progression except for the sixth term 89 translates into
{{{S-a[6]=99}}}
{{{system(S-a[1]=99,S-a[6]=89)}}} --> {{{a[6]-a[1]=99-89}}} --> {{{a[6]-a[1]=10}}}
 
The nth term in an arithmetic progression is given by the formula
{{{a[n]=a[1]+(n-1)*d}}} so
{{{a[6]=a[1]+5d}}} --> {{{a[6]-a[1]=5d}}}
 
{{{system(a[6]-a[1]=10,a[6]-a[1]=5d)}}} --> {{{5d=10}}} --> {{{d=2}}}
 
We are told that the sum of the first term and the fifth term is equal to 10.
The fifth term is
{{{a[5]=a[1]+(5-1)*2}}} --> {{{a[5]=a[1]+8}}}
The sum of the first term and the fifth term is
{{{a[1]+(a[1]+8)=10}}} --> {{{2a[1]+8=10}}} --> {{{2a[1]=10-8}}} --> {{{2a[1]=2}}} --> {{{a[1]=1}}}
 
Knowing that the first term is {{{a[1]=1}}} and the common difference is {{{d=2}}} we can calculate the third term as
{{{a[3]=1+(3-1)*2}}} --> {{{a[3]=1+2*2}}} --> {{{a[3]=1+4}}} --> {{{highlight(a[3]=5)}}}