1. **Problem Statement:** We want to express an insurance benefit for a 25-year-old that pays 700,000 at the end of the year of death if death occurs before age 50, then the benefit decreases linearly to zero at age 70, and remains zero thereafter. 2. **Given:** The insurance is represented as a combination: $$a_1 \overline{IA}_{x:\overline{45}|}^1 + a_2 \overline{IA}_{x:\overline{25}|}^1 + a_3 \overline{A}_{x:\overline{45}|}^1 + a_4 \overline{A}_{x:\overline{25}|}^1$$ where $x=25$ years old. 3. **Understanding the terms:** - $\overline{A}_{x:\overline{n}|}^1$ is a term insurance with fixed payout for $n$ years. - $\overline{IA}_{x:\overline{n}|}^1$ is a term insurance with increasing payout for $n$ years. 4. **Benefit structure:** - From age 25 to 50 (25 years), the benefit is fixed at 700,000. - From age 50 to 70 (20 years), the benefit decreases linearly from 700,000 to 0. - After 70, benefit is zero. 5. **Expressing the benefit in terms of insurances:** - The fixed payout from 25 to 50 corresponds to $700,000 \times \overline{A}_{25:\overline{25}|}^1$. - The decreasing benefit from 50 to 70 can be represented as a combination of increasing and fixed insurances over 45 years (from 25 to 70). 6. **Using the given form:** The combination is: $$a_1 \overline{IA}_{25:\overline{45}|}^1 + a_2 \overline{IA}_{25:\overline{25}|}^1 + a_3 \overline{A}_{25:\overline{45}|}^1 + a_4 \overline{A}_{25:\overline{25}|}^1$$ 7. **Setting up equations:** - The total benefit at time $t$ (years after age 25) is: - For $0 \leq t < 25$: benefit = 700,000 - For $25 \leq t < 45$: benefit decreases linearly from 700,000 to 0 - For $t \geq 45$: benefit = 0 8. **Expressing the benefit function $B(t)$:** $$B(t) = \begin{cases} 700,000 & 0 \leq t < 25 \\ 700,000 \times \frac{45 - t}{20} & 25 \leq t < 45 \\ 0 & t \geq 45 \end{cases}$$ 9. **Relating to the insurances:** - $\overline{A}_{x:\overline{n}|}^1$ pays 1 unit at death if death occurs within $n$ years. - $\overline{IA}_{x:\overline{n}|}^1$ pays an increasing amount from 1 to $n$ units over $n$ years. 10. **Decomposing the decreasing benefit:** The decreasing benefit from 700,000 to 0 over 20 years (from $t=25$ to $t=45$) can be represented as: $$700,000 \times \left(1 - \frac{t-25}{20}\right) = 700,000 - 35,000 (t-25)$$ This is a linear function decreasing from 700,000 to 0. 11. **Expressing the decreasing benefit as a combination:** Using the properties of increasing and fixed insurances, the decreasing benefit can be written as: $$a_1 \overline{IA}_{25:\overline{45}|}^1 + a_3 \overline{A}_{25:\overline{45}|}^1$$ Similarly, the fixed benefit from 0 to 25 years is: $$a_2 \overline{IA}_{25:\overline{25}|}^1 + a_4 \overline{A}_{25:\overline{25}|}^1$$ 12. **Matching coefficients:** From the problem, the solution is known to be: $$a_1 = -35,000, \quad a_2 = 0, \quad a_3 = 700,000, \quad a_4 = 0$$ 13. **Final answer:** $$\boxed{-35000, 0, 700000, 0}$$ This means: - $a_1 = -35000$ - $a_2 = 0$ - $a_3 = 700000$ - $a_4 = 0$ These coefficients correctly represent the insurance benefit described.