Write as sum: $2x_{1}x_{2}+3x_{2}x_{3}^{2}+4x_{3}x_{4}^{3}+5x_{4}x_{5}^{4}+6x_{5}x_{6}^{5}$

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Write as sum: $2x_{1}x_{2}+3x_{2}x_{3}^{2}+4x_{3}x_{4}^{3}+5x_{4}x_{5}^{4}+6x_{5}x_{6}^{5}$

I wrote like this but not sure if correct. I made tests and it seems correct but I ask just to be very sure:

$$\sum_{i=1}^{5}(i+1)x_{i}x_{i+1}^{i}$$

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I think you want $$ \sum_{i=1}^{5}(i+1)x_{i}x_{i+1}^{i}. $$