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Prove that the function computing the product ( 1 2 + 1 ) * ( 2 2 + 1 ) * ( 3 2 +

Prove that the function computing the product
(12+1)*(22+1)*(32+1)*...*(n2+1)
is primitive recursive. This proof should follow the same pattern that we used in class to prove that addition and multiplication are primitive recursive:
You start with a 3-dot expression
First you write a for-loop corresponding to this function
Then you describe this for-loop in mathematical terms
Then, to prepare for a match with the general expression for primitive recursion, you rename the function to f and the parameters to n1,..., m
Then you write down the general expression of primitive recursion for the corresponding k
Then you match: find g and h for which the specific case of primitive recursion will be exactly the functions corresponding to initialization and to what is happening inside the loop
Then, you get a final expression for the function
(12+1)*(22+1)*(32+1)*...*(n2+1)
that proves that this function is primitive recursive, i.e., that it can be formed from 0,\pi ki, and \sigma by composition and primitive recursion.

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