You can use the formula you suggested
=PRODUCT(A1:A10+5)
....but that's an "array formula" so you have to enter with CTRL+SHIFT+ENTER key combination
Put formula in cell then select that cell and press F2 to select formula. Now hold down CTRL and SHIFT while presssing ENTER. If done correctly you will see curly braces around the formula so it looks like this:
{=PRODUCT(A1:A10+5)}
to avoid "array entry" you can add an INDEX
function, i.e. this regular version
=PRODUCT(INDEX(A1:A10+5,0))