Question 979908


Let me to decipher the abbreviation &nbsp;<B>AP</B>. &nbsp;It is &nbsp; "Arithmetic Progression".

 
Let &nbsp;<B>x</B>&nbsp; be the &nbsp;<B>third</B>&nbsp; term of our arithmetic progression and &nbsp;<B>d</B>&nbsp; be its &nbsp;<B>common difference</B>. 

Then the five first terms are


x - 2d
x - d
x
x + d
x + 2d.


The sum of these terms is &nbsp;5x. &nbsp;The sum of their squares is {{{5x^2}}} + {{{10d^2}}}. &nbsp;It is easy to check.


Thus we have the system of two &nbsp;(non-linear)&nbsp; equations


{{{system(5x = 10,
5x^2 + 10d^2 = 380)}}}.


From the first equation we have &nbsp;x = {{{10/5}}} = 2.

Substitute it into the second equation. &nbsp;You will get 


{{{5*2^2}}} + {{{10d^2}}} = {{{380}}}, &nbsp;&nbsp;or&nbsp;&nbsp; {{{10d^2}}} = {{{380 - 20}}} = {{{360}}}.


Hence, &nbsp;{{{d^2}}} = {{{360/10}}} = 36 &nbsp;and&nbsp; d = +/-6.


Therefore, &nbsp;the members of the progression are 

-10, -4, 2, 8, 14


- in this order or in the opposite order - it doesn't matter.


You can check that this progression &nbsp;(or these two progressions)&nbsp; satisfy to the given conditions.