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I'd like to know how to create a pattern for these sum:

1) $a_{1}a_{2}a_3+a_1a_2a_4+...+a_1a_{n-1}a_n+a_2a_3a_4+a_2a_3a_5+...+ a_2a_{n-1}a_{n}+ ...+a_{n-2}a_{n-1}a_n = ?$

2) $a_{1}a_{2}a_3a_4+a_1a_2a_4a_5+...+a_1a_{n-2}a_{n-1}a_n+a_2a_3a_4a_5+a_2a_3a_5a_6+...+ a_2a_{n-2}a_{n-1}a_{n}+ ...+a_{n-3}a_{n-2}a_{n-1}a_n = ?$

It may be analogically like for $a_1a_2+...+a_1a_n+...+a_{n-1}a_n=\sum\sum_{1 \le i <j \le n} a_ia_j$

Gregor
  • 708

1 Answers1

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1) $\sum^{n-2}_{i=1}a_i(\sum^{n-1}_{j=i+1}a_j(\sum^n_{k=j+1}a_k))$

or $\sum\sum\sum_{1 \le i<j<k\le n }a_ia_ja_k$

2)$\sum^{n-3}_{i=1}a_i(\sum^{n-2}_{j=i+1}a_j(\sum^{n-1}_{k=j+1}a_k(\sum^n_{l=k+1}a_l)))$

or $\sum\sum\sum\sum_{1 \le i<j<k <l\le n }a_ia_ja_ka_l$

sasas2
  • 58