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To solve this problem, we can use the formula for compound interest to find the present value (the initial lump sum) that Tanja must invest. The compound interest formula is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- $ A $ is the amount of money desired in the future ($200,000).
- $ P $ is the initial principal or lump sum (the amount Tanja needs to invest today).
- $ r $ is the annual interest rate (6% or 0.06).
- $ n $ is the number of times interest is compounded per year (quarterly, so $ n = 4 $).
- $ t $ is the number of years the money is invested (15 years).

To find $ P $, we rearrange the formula:

\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]

### Plugging in the values:

- $ A = 200,000 $
- $ r = 0.06 $
- $ n = 4 $
- $ t = 15 $

### Calculation:

\[
P = \frac{200,000}{\left(1 + \frac{0.06}{4}\right)^{4 \cdot 15}}
\]

1. Calculate the quarterly interest rate: $ \frac{0.06}{4} = 0.015 $.
2. Calculate the exponent $ 4 \times 15 = 60 $.
3. Calculate $ \left(1 + 0.015\right)^{60} $.
4. Finally, divide $ 200,000 $ by this result to get $ P $.

Let me calculate this for you.Tanja must invest approximately **$81,859.19** today to have $200,000 available at retirement in 15 years, given a 6% interest rate compounded quarterly.

Would you like more details on the calculations, or do you have any other questions?

Here are some related questions for further exploration:
1. What would the investment amount be if the interest was compounded monthly instead?
2. How would the initial investment change if Tanja needed the money in 10 years instead of 15?
3. What if the interest rate were lower, say 4%? How much would Tanja need to invest?
4. How does quarterly compounding compare with annual compounding in terms of returns over 15 years?
5. How much more would Tanja have if she invested an extra $1,000 today?

**Tip:** Compound interest calculations are highly sensitive to changes in the interest rate and compounding frequency. Small differences can significantly impact the final amount.

Tanja wants to establish an account that will supplement her retirement income beginning 15 years from now. Find the lump sum she must deposit today so that $200,000 will be available at time of retirement, if the interest rate is 6%, compounded quarterly.

Learn how to calculate the initial lump sum Tanja must invest today to reach $200,000 in 15 years at a 6% interest rate, compounded quarterly. This problem involves compound interest and present value calculations, suitable for high school students and financial math learners.

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