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Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is

$$\triangledown a_{n} = a_{n} − a_{n−1}.$$

The $(k + 1)^{\text{st}}$ difference $\triangledown^{k+1}a_{n}$ is obtained from $\triangledown ^{k} a_{n}$ by

$$\triangledown ^{k+1}a_{n} = \triangledown^{k}a_{n} − \triangledown ^{k}a_{n−1}.$$

Find $\triangledown^{2} a_{n}$ for the sequence $\{a_{n}\},$ where

  1. $a_{n} = 4.$
  2. $a_{n} = 2n.$
  3. $a_{n} = n^{2}.$
  4. $a_{n} = 2^{n}.$
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