To solve the problem using the geometric progression formula, we can take advantage of the fact that the annuity payments increase at a constant rate (2.5% per annum). This allows us to express the sum of the present values of the annuity payments as a geometric series.

### Rewriting the Formula

The expected present value (EPV) of the annuity can be rewritten as a geometric series:

\[
EPV = \sum_{t=1}^{\infty} 2000 \times (1.025)^{t-1} \times v^t \times \, _tp_{68}
\]

Where:
- \( v = \frac{1}{1.066} \) is the discount factor.
- \( \, _tp_{68} \) is the probability of survival to time \( t \) from age 68.
- \( 2000 \times (1.025)^{t-1} \) is the payment amount at time \( t \).

Let's denote:
- \( A_1 = 2000 \) (the initial payment),
- \( g = 1.025 \) (the growth factor),
- \( v = 0.9379 \) (discount factor),
- \( r = v \times g \) (the combined discount and growth factor).

Then the EPV can be expressed as:

\[
EPV = A_1 \sum_{t=1}^{\infty} \left(v^t \times (1.025)^{t-1} \times \, _tp_{68}\right)
\]

This series is geometric with the first term \( a = 2000 \times v \) and common ratio \( r = v \times g \).

### Simplifying Using the Geometric Series Formula

The sum of an infinite geometric series with the first term \( a \) and common ratio \( r \) (where \( |r| < 1 \)) is given by:

\[
\text{Sum} = \frac{a}{1 - r}
\]

Thus, the EPV becomes:

\[
EPV = 2000 \times \sum_{t=1}^{\infty} \left(\frac{(1.025)^{t-1}}{(1.066)^t} \times \, _tp_{68}\right)
\]

This can be expressed as:

\[
EPV = 2000 \times \sum_{t=1}^{\infty} \left(\frac{1}{1.066} \times \frac{(1.025)^{t-1}}{(1.066)^{t-1}} \times \, _tp_{68}\right)
\]

Given \( r = \frac{1.025}{1.066} = \frac{g}{1+i} \):

\[
EPV = 2000 \times \sum_{t=1}^{\infty} \left(v^t \times g^{t-1} \times \, _tp_{68}\right)
\]

Now, simplifying the geometric series:

\[
EPV = 2000 \times \frac{1}{1 - r} \times \sum_{t=1}^{\infty} \left(\, _tp_{68}\right)
\]

This simplifies to:

\[
EPV = 2000 \times \frac{1}{1 - \frac{1.025}{1.066}} \times \sum_{t=1}^{\infty} \left(\, _tp_{68}\right)
\]

Finally, the EPV becomes:

\[
EPV = 2000 \times \frac{1.066}{1.066 - 1.025} \times \sum_{t=1}^{\infty} \left(\, _tp_{68}\right)
\]

\[
EPV = 2000 \times \frac{1.066}{0.041} \times \sum_{t=1}^{\infty} \left(\, _tp_{68}\right)
\]

### Final Calculation

Now, assuming that you have the mortality probabilities \( \, _tp_{68} \) (from the PFA92C20 table), you can substitute these values into the formula and calculate the final EPV.

Would you like to proceed with specific calculations, or would you like help with any particular part of the process?

Explore how to use the geometric progression formula to calculate the expected present value (EPV) of annuity payments increasing at a constant rate. Learn about geometric series and discount factors applied in finance. Ideal for advanced learners and professionals.

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