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Q$3.$Depicted in the picture below is the $($not to scale$)$ graph of the size of the continuous insurance benefits for a policy of term $21$ years.

$2,1$

The shape of the benefit can be constructed using policies involving variations of insurance benefits that are constant $($like $\frac{?}{\text{b}\text{a}\text{r}}(A{)}_x$ for instance$)$ and linearly increasing $($like $\frac{?}{\text{b}\text{a}\text{r}}(IA{)}_x).$

$($a$)$ One such combination includes the use of deferred insurance, and can be written in the form

$\frac{a_1}{b}ar(IA{)}_{x\frac{\text{:}}{b}ar(4)|}^1+\frac{a_2}{b}ar(A{)}_{x\frac{\text{:}}{b}ar(4)|}^1+\frac{a_{34|}}{b}ar(A{)}_{x\frac{\text{:}}{b}ar(11)}^1+\frac{a_{415|}}{b}ar(A{)}_{x\frac{\text{:}}{b}ar(6)|}^1$

Enter the values of $a_1,a_2,a_3,$ and $a_4($in that order$)$ into the answer box below, separated with commas.

$($b$)$ Another combination does not include any deferred insurance, and can be written in the form

$\frac{b_1}{b}ar(A{)}_{x\frac{\text{:}}{b}ar(21)}^1+\frac{b_2}{b}ar(A{)}_{x\frac{\text{:}}{b}ar(15)}^1+\frac{b_3}{b}ar(A{)}_{x\frac{\text{:}}{b}ar(4)|}^1+\frac{b_4}{b}ar(IA{)}_{x\frac{\text{:}}{b}ar(4)}^1$

Enter the values of $b_1,b_2,b_3,$ and $b_4($in that order$)$ into the answer box below, separated with commas.

Note that the required constants can potentially be negative. Q$4.$Similar to the previous question, a combination of term insurance involving fixed payouts $($A$)$ and increasing payouts $($IA$)$ can be used to provide a $25-$year$-$old person insurance that pays $$600,000$ at the end of the year of death if death occurs before the age of $40,$ but for which the benefit then decreases linearly down to a benefit of zero after turning $60.$ The benefit then remains at zero thereafter. Such a combination that does not use deferred insurance is of the form

a$1$IA$\_\_1$

x: $35$

$+$a$2$IA$\_\_1$

x: $15$

$+$a$3$A$\_1$

x: $35$

$+$a$4$A$\_1$

x: $15$

$.$

Enter the values of a$1,$ a$2,$ a$3,$ and a$4($in that order$)$ into the answer box below, separated with commas.

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