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The formula for degrees of unsaturation is:

$$1+ \frac{1}{2} \sum ( n_i (v_i-2))$$

$v_i$ = valency of atom
$n_i$ = number of atoms with that valency

Could someone explain how the formula works, and why divalent atoms don't contribute to unsaturation (as evident from the formula).

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This is the correct formula:

$$ DU = 1 + \frac{1}{2}\sum _{v=1}^{j} n_{i}(v-2),$$ where

$j$ = maximum valence of any atom in molecule
$v$ = valence of atom(s) 1 to whatever
$n_{v}$ = number of atoms with that valence

Wikipedia gives a good explanation. In simple terms the formula will give the number of double bonds in an alkene. For all alkanes the number would be 0 of course. Rings and ethynes would complicate the use of the formula. Think of using the equation on chemical formulas like $\ce{C_{x}}\ce{H_{y}}$ or $\ce{C_{x}}\ce{H_{y}}\ce{O_{z}}$.

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