An $n$ year annuity which pays $k^2$ at the end of the $k^{th}$ year for $k=1,2,3,4,$dots,$n-1,n$ has a present value of $x_n.$

$($a$)$ Show that

$|)-n^2{v}^n$

$($b$)$ Let $x$ be the present value for a perpetuity which pays $k^2$ at the end of the $k^{th}$ year for $k=1,2,3,4,$dots. Show that

$x=\frac{1}{d}(\frac{1}{i}+\frac{2}{i^2})$

$($c$)$ Mr$.$ Seven makes $10$ end of year deposits into a savings account which credits interest at an annual effective interest rate of $5\%.$ The annual deposits $($in sequential order$)$ are $1000,4000,9000,16,000,25,000,36,000,49,000,64,000,81,000$ and $100,000.$ Find the accumulated value of the $10$ annual deposits just after the final deposit.

$($d$)$ Refer to $($c$).$ Mr$.$ Seven has decided to use the accumulated value of the $10$ annual deposits to purchase a perpetuity immediate. The first payment for the perpetuity will be exactly one year after the $10$ th and final deposit. The payments go as follows. $Y,4Y,9Y,16Y,25Y,36Y,49Y,64Y,$dots. The present value for the perpetuity is based on an annual effective interest rate of $0\mathrm{.}5\%.$ Find the maximum value for $Y.$