You can try to show this by induction:
• According to the given closed form, we have
, which agrees with the initial value <em>S</em>₁ = 1.
• Assume the closed form is correct for all <em>n</em> up to <em>n</em> = <em>k</em>. In particular, we assume

and

We want to then use this assumption to show the closed form is correct for <em>n</em> = <em>k</em> + 1, or

From the given recurrence, we know

so that






which is what we needed. QED