Question 87738
Write a rational expression, in its simplest form, that represents the sum of the reciprocals of the squares of the consecutive even integers n-2, n, and n+2. That is the expression should be in terms of the integer variable n.
:
Assume this problem means:
{{{1/((n-2)^2)}}} + {{{1/n^2}}} + {{{1/((n+2)^2)}}}
:
Place the expression over a single denominator:
{{{((n^2(n+2)^2) + ((n-2)^2*(n+2)^2) + (n^2(n-2)^2))/n^2(n-2)^2(n+2)^2)}}}
:
FOIL all those squares:
{{{(n^2(n^2+4n+4) + (n^2-4n+4)(n^2+4n+4) + n^2(n^2-4n+4))/(n^2(n^2-4n+4)(n^2+4n+4))}}}
:
Multiply (n^2-4n+4)(n^2+4n+4) the vertical way, you should get:
{{{n^4 - 8n^2 + 16 }}}; then we have:
{{{((n^4+4n^3+4n^2) + (n^4-8n^2+16) + (n^4-4n^3+4n^2))/(n^2(n^4-8n^2+16))}}}
:
Group like terms in the numerator and mult in the denominator:
{{{(n^4+n^4+n^4+4n^3-4n^3+4n^2-8n^2+4n^2+16)/(n^6-8n^4+16n^2)}}} =
:
{{{(3n^4+16)/(n^6-8n^4+16n^2)}}} is what I get here, hope this helps you.