Problem $1.$ An insurance company is offering a new policy for its customers. Typically the policy is bought by a parent or grandparent for a child at the child birth. The details of the policy are as follows: The purchaser $($say$,$ the parent$)$ makes the following six payments to the insurance company:

First birthday: $,$$$6,000$

Second birthday: $$7,000$

Third birthday: $$8,000$

Fourth birthday: $$9,000$

Fifth birthday: $,$$$10,000$

Sixth birthday: $$11,000$

After the child's sixth birthday, no more payments are made. When the child reaches age $65,$ he or she receives $$4,500,000.$ If the relevant annual interest rate is $6$ percent for the first six years and $8$ percent for all subsequent years, is the policy worth buying? Show you work. Ignore transaction costs and taxes for the problem.