$(4)$ Now consider the second altemative$-4$ annual payments of $ $15,000$ each. Assume that the payments are made at the end of each year.

$($a$)$ What type of annuity is this?

$($b$)$ What is the future value of this annuity if the payments are invested in an account paying $6$ percent interest annually?

$($c$)$ What size payment would be needed to accumulate $$140,000$ under annual compounding at a $6$ percent interest rate?

$($d$)$ What lump sum deposited today would produce the same value as in part $($b$)$

$(5)$ Repeat the above analysis assuming the payments are made at the beginning of the period.

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