Taking the lump-sum($50 million before income taxes) or annuity(an initial payment and then 29 annual payments that increase by 5% each year for the next 29 years). How much is the annual payment per year and the total sum if the initial payment is $1.5 million.

To compare the lump-sum option of $50 million before taxes with the annuity option, let’s first calculate the total value of the annuity over the 30 years, including the initial payment.

### Given:
- Initial annuity payment (Year 1): \$1.5 million
- Annual increase: 5%
- Number of years: 30 years (including the first payment)
  
The annuity is a growing annuity, where each year’s payment increases by 5%. The formula for the total value of a growing annuity is:

\[
A_{\text{total}} = P \times \left[ \frac{1 - (1 + g)^{-n}}{r - g} \right]
\]

Where:
- $A_{\text{total}}$ is the total value of the annuity over 30 years
- $P$ is the initial payment (\$1.5 million)
- $g$ is the growth rate (5% or 0.05)
- $n$ is the number of years (30)
- $r$ is the discount rate (if you want the present value of future payments, which isn't mentioned here)

For simplicity, let's calculate the total sum of the payments without discounting future values (i.e., assuming the payments are not being discounted to present value).

---

### Step 1: Calculate the payment for each year
The payment each year increases by 5% of the previous year's payment. The formula for the payment in year $n$ is:

\[
A_n = A_1 \times (1 + g)^{n-1}
\]

Where:
- $A_n$ is the payment in year $n$
- $A_1 = 1.5 \text{ million}$ is the initial payment
- $g = 0.05$ is the annual growth rate

### Step 2: Calculate the total sum of the annuity payments over 30 years
The total sum of the payments is the sum of all the individual payments over 30 years:

\[
A_{\text{total}} = A_1 + A_2 + A_3 + \dots + A_{30}
\]

We can also express this as the sum of a geometric progression:

\[
A_{\text{total}} = A_1 \times \left( 1 + (1 + g) + (1 + g)^2 + \dots + (1 + g)^{29} \right)
\]

I will now calculate the total sum of the annuity payments over 30 years.The total sum of the annuity payments over 30 years, with an initial payment of \$1.5 million and a 5% annual increase, would be approximately **\$99.66 million**.

This compares with the lump-sum option of \$50 million before taxes.

### Key breakdown:
- **First year's payment**: \$1.5 million.
- **Each subsequent payment increases by 5%** annually for 29 years.
  
Would you like a detailed calculation for each year's payment, or any other clarifications?

Here are 5 related questions you might explore:
1. What is the payment amount in the final (30th) year?
2. How does the lump-sum compare after applying taxes?
3. What is the present value of the annuity, assuming a certain discount rate?
4. How do changes in the growth rate affect the total annuity?
5. What happens if the initial payment is higher or lower?

**Tip**: Always consider the tax implications when comparing lump-sum and annuity options!

Compare a lump-sum of $50 million before taxes versus an annuity starting at $1.5 million with a 5% annual increase over 30 years. Learn how to calculate the total sum of payments using the geometric series formula. Suitable for high school or college-level students.

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