Question 974831
The sum of the following is 1 + 3 - 5 + 7 + 9 - 11+ 13......... 3n terms
<pre>
The given sequence is the sum of two arithmetic sequences, 3n terms of
the first sequence and n terms of the second sequence:

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + ... + 3nth term
      -10         - 22           - 34           - 46 - ... -  nth term
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1 + 3 - 5 + 7 + 9 - 11 + 13 + 15 - 17 + 19 + 21 - 23 + ...     ?

The first sequence has 3n terms, but first we find the sum of n terms 

{{{S[n]}}}{{{""=""}}}{{{expr(n/2)*(2a[1]^""+(n-1)d^"")}}}{{{""=""}}}{{{expr(n/2)*(2(1)^""+(n-1)2^"")}}}{{{""=""}}}{{{expr(n/2)*(2+2n-2)}}}{{{""=""}}}{{{expr(n/2)*(2n)}}}{{{""=""}}}{{{n^2}}}

and since it has 3n terms, we replace n by 3n and get the sum of 3n terms,
{{{(3n)^2}}} or {{{9n^2}}} 

The second sequence -10-22-34-46- ... has only n terms:

{{{S[n]}}}{{{""=""}}}{{{expr(n/2)*(2a[1]^""+(n-1)d^"")}}}{{{""=""}}}{{{expr(n/2)*(2(-10)^""+(n-1)(-12)^"")}}}{{{""=""}}}{{{expr(n/2)*(-20-12n+12)}}}{{{""=""}}}
{{{expr(n/2)*(-8-12n)}}}{{{""=""}}}{{{expr(n/2)*2(-4-6n)}}}{{{""=""}}}{{{n*(-4-6n)}}}{{{""=""}}}{{{-4n-6n^2}}}

Adding the two sums:  {{{9n^2-4n-6n^2}}}{{{""=""}}}{{{3n^2-4n}}}

Answer: {{{3n^2-4n}}}
 
Edwin</pre>