Jenny will invest $$1,000,000$ in an account.

$2\mathrm{.}1.$ What is the future value of her investment in $20$ years if the bank offers an annual percentage rate of $7\%$:

a$.$ Compounded annually $($assuming that the bank$$s payments are in the form of an annuity due$).$

b$.$ Compounded quarterly $($assuming that the bank$$s payments are in the form of an annuity due$).$

c$.$ Compounded monthly $($assuming that the bank$$s payments are in the form of an ordinary annuity$).$

d$.$ Compounded continuously $($assuming that the bank$$s payments are in the form of an ordinary annuity$).$

$2\mathrm{.}2.$ Why does the future value increase as the compounding period shortens?