Prove that the function computing the product

$(12+1)*(22+1)*(32+1)*...*($n$2+1)$

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 n$1,...,$ 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

$(12+1)*(22+1)*(32+1)*...*($n$2+1)$

that proves that this function is primitive recursive, i$.$e$.,$ that it can be formed from $0,\backslash$pi ki$,$ and $\backslash$sigma by composition and primitive recursion.