(a) What amount did she save during the nineteenth month?

Formula:

The monthly savings form a geometric sequence. The $n$-th term of a geometric sequence is given by: $a_n = a \cdot r^{n-1}$ Where:

• $a$: the first term
• $r$: the common ratio
• $n$: the term number

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Stocktake:

$a = 400, \quad r = 1.11, \quad n = 19$

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Substitute:

$a_{19} = 400 \cdot (1.11)^{19-1}$

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Solve:

$a_{19} = 400 \cdot (1.11)^{18}$

Using a calculator: $(1.11)^{18} \approx 6.143$ $a_{19} \approx 400 \cdot 6.143 = 2457.2$

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Final Answer:

Janet saved R2457.20 in the nineteenth month.

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(b) How much did she save all together in her first 17 working months?

Formula:

The sum of the first $n$ terms of a geometric sequence is: S_n = a \cdot \frac{1-r^n}{1-r} \quad \text{(for r > 1)}

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Stocktake:

$a = 400, \quad r = 1.11, \quad n = 17$

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Substitute:

$S_{17} = 400 \cdot \frac{1 - (1.11)^{17}}{1 - 1.11}$

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Solve:

$(1.11)^{17} \approx 5.534$ $S_{17} = 400 \cdot \frac{1 - 5.534}{-0.11}$ $S_{17} = 400 \cdot \frac{-4.534}{-0.11}$ $S_{17} = 400 \cdot 41.22 \approx 16488$

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Final Answer:

Janet saved R16,488 in her first 17 months.

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(c) In which month of her working life was her total savings more than R20,100 for the first time?

Formula:

We solve for $n$ using the sum formula for a geometric sequence: $S_n = a \cdot \frac{1 - r^n}{1 - r}$ Rearranging to find $n$, we solve: $\frac{S_n \cdot (1 - r)}{a} + 1 = r^n$

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Stocktake:

$S_n = 20100, \quad a = 400, \quad r = 1.11$

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Substitute:

$\frac{20100 \cdot (1 - 1.11)}{400} + 1 = (1.11)^n$ $\frac{20100 \cdot -0.11}{400} + 1 = (1.11)^n$ $-552.75 + 1 = (1.11)^n$ $552.75 = (1.11)^n$

Taking the logarithm: $n = \frac{\log(552.75)}{\log(1.11)}$

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Solve:

Using a calculator: $\log(552.75) \approx 2.743, \quad \log(1.11) \approx 0.04576$ $n = \frac{2.743}{0.04576} \approx 59.92$

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Final Answer:

Janet's total savings exceeded R20,100 during the 60th month.

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Summary:

• (a) Janet saved R2457.20 in the 19th month.
• (b) Janet saved R16,488 in the first 17 months.
• (c) Janet’s total savings exceeded R20,100 for the first time in the 60th month.

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Would you like further clarification or detailed breakdowns? 😊

Here are five related questions for exploration:

1. How would her savings pattern change if the growth rate was reduced to 8% instead of 11%?
2. How much would she save in total after 24 months?
3. What is the average monthly savings over the first 17 months?
4. How long would it take for her to save a total of R50,000?
5. What is the percentage increase in savings from the 19th month to the 20th month?

Tip: For geometric growth problems, logarithms are essential when solving for the term count!