Question 3
Calculate the accumulated amount after eight years of weekly payments of R1 900 each into an account
earning interest per year, compounded weekly.

Question

Question 3 Calculate the accumulated amount after eight years of weekly payments of R1 900 each into an account earning $$ 9,7 \% $$ interest per year, compounded weekly.

UpStudy AI · Accepted Answer

We are given:

- Weekly payment (annuity payment): $$ P = 1900 $$  
- Annual nominal interest rate: $$ r = 9.7\% = 0.097 $$  
- Compounding (and payment) frequency: 52 times per year  
- Total number of payments: $$ n = 8 \times 52 = 416 $$

Step 1. Calculate the periodic (weekly) interest rate:

$$
i = \frac{r}{52} = \frac{0.097}{52} \approx 0.00186538
$$

Step 2. Use the formula for the accumulated amount of an annuity:

$$
A = P \times \frac{(1+i)^n - 1}{i}
$$

Substitute the values:

$$
A = 1900 \times \frac{(1+0.00186538)^{416} - 1}{0.00186538}
$$

Step 3. Compute $$(1+i)^{416}$$:

$$
(1+0.00186538)^{416} \approx \exp\Bigl(416 \times \ln(1.00186538)\Bigr)
$$

Approximating the natural logarithm (since for small $$i$$, $$\ln(1+i) \approx i$$):

$$
\ln(1.00186538) \approx 0.00186538,
$$
$$
416 \times 0.00186538 \approx 0.775.
$$

Thus,

$$
(1+0.00186538)^{416} \approx \exp(0.775) \approx 2.170.
$$

Step 4. Substitute back into the annuity formula:

$$
A \approx 1900 \times \frac{2.170 - 1}{0.00186538}
$$
$$
A \approx 1900 \times \frac{1.170}{0.00186538}
$$

Step 5. Compute the annuity factor:

$$
\frac{1.170}{0.00186538} \approx 627.69.
$$

Step 6. Finally, compute the accumulated amount:

$$
A \approx 1900 \times 627.69 \approx 1,\!192,\!611.
$$

Thus, the accumulated amount after eight years is approximately $$ R1,\!192,\!611 $$.
The accumulated amount after eight years is approximately R1,192,611.

Question 3
Calculate the accumulated amount after eight years of weekly payments of R1 900 each into an account
earning interest per year, compounded weekly.

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Question 3 Calculate the accumulated amount after eight years of weekly payments of R1 900 each into an account earning interest per year, compounded weekly.

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Question 3
Calculate the accumulated amount after eight years of weekly payments of R1 900 each into an account
earning interest per year, compounded weekly.