Question 98709
Sequence is just some (ordered) list of numbers(terms).The equation {{{A[n] = n^3-1}}} just tells you how to get n-th term. If you want to get first term, substitute 1 for n, so you get {{{A[1] = 1^3-1 = 1-1 = 0}}} , same with second term : substitute 2, in order to get : {{{A[2] = 2^3-1 = 8-1 = 7}}}. Similiarly with {{{A[n] = (n-1)^2}}}
1st : {{{A[1] = (1-1)^2 = 0^2 = 0}}}
2nd : {{{A[2] = (2-1)^2 = 1^2 = 1}}}
3rd : {{{A[3] = (3-1)^2 = 2^2 = 4}}}
and so on...

Summation- you are just summing terms of sequence. For instance, if you have : 
{{{sum(n^3-1,n=1,6)}}}
 , same like in the first problem, !but summed! So not 0 , 7 , 25 , 63 , but 0+7+25+63 = 95.(You are substituting through n=1 to n=6 and summing together.

or {{{sum( n , n=0 ,5)}}} = 0+1+2+3+4+5 = 15
given 4+8+12+16+20 you can rewrite as {{{sum(4*n,n = 1 , 5)}}}
-7-8-9-10-10-11 as {{{sum(-n,n=7, 11)}}}
and finally 5/6+6/7+7/8+8/9+... as {{{sum(n/(n+1),n = 5 , infinity)}}}
  
-method of rewriting into summation notation is always based on some preliminary thought, as multiples of 4 etc.. so go through it and give some thought about it...