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For[[[[[[[[[[[Do[]]]]]]]]]]]]]

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Answering the question in the title:

n = 8;
r = 6;
i = 3;

Array[t, r, i, f] == Array[t, r - 1, i + 1, f] - Array[t, r - 1, i, f]

(*    f[t[3], t[4], t[5], t[6], t[7], t[8]] ==
      f[t[4], t[5], t[6], t[7], t[8]] - f[t[3], t[4], t[5], t[6], t[7]]    *)

For your code, better use t[i] instead of $t_i$ subscripts.

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