Question 1047950
{{{S[5] = 25 = 2a[1]+4d}}}  ===> {{{a[1] = 5-2d}}}


===> {{{a[1]^2 + a[2]^2 + a[3]^2  + a[4]^2 + a[5]^2  = 165}}}

<===> {{{(5-2d)^2 + (5-d)^2 +5^2 + (5+d)^2 + (5+2d)^2 = 165}}}

After expanding each term accordingly, you should get

{{{125 + 10d^2 = 165}}}   

===> {{{d^2 = 4}}}

===> d = 2 or -2.

(If d = 2, the sequence is 1,3,5,7,9.  If d = -2, the sequence is 9,7,5,3,1.  These are two different sequences, both of which 
satisfy the initial conditions.)