Question Statement

Sara pays an insurance company for her vehicle in installments.

• Her first installment is Rs. 8,000.
• Each subsequent installment increases by 5%.

What is the total amount she will pay in 24 installments?

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Background and Explanation

This problem describes payments that increase by a constant percentage, which is a hallmark of a geometric sequence. Each payment is a term in the sequence, and the constant 5% increase determines the common ratio ($r$). To find the total amount paid, we must calculate the sum of the first 24 terms of this sequence, which is known as a geometric series.

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Solution

To find the total amount Sara pays, we need to sum all 24 of her installments.

1. Identify the Geometric Sequence:

   • The first installment is the first term: $a_1=8000$.
   • Each installment increases by 5%. This means the next payment is 100% of the previous payment plus an extra 5%, which is 105% of the previous amount.
   • The common ratio ($r$) is therefore $105\%$, or $1.05$.
   • The number of installments is $n=24$.

2. Confirm the Pattern:

   • $1^{st}$ installment: $a_1=8000$
   • $2^{nd}$ installment: $a_2=8000\times (1.05)$
   • $3^{rd}$ installment: $a_3=8000\times (1.05{)}^2$

3. Calculate the Total Amount Paid ($S_{24}$): We need to find the sum of the first 24 installments. We use the formula for the sum of a finite geometric series.

   The formula is: $S_n=\frac{a_1(r^n-1)}{r-1}$

   Substitute the known values ($a_1=8000$, $r=1.05$, and $n=24$) into the formula:

   $S_{24}=\frac{8000((1.05{)}^{24}-1)}{1.05-1}$

   First, calculate $(1.05{)}^{24}$: $(1.05{)}^{24}\approx 3.2251$

   Now, substitute this back into the formula:

   $S_{24}\approx \frac{8000(3.2251-1)}{0.05}$

   $S_{24}\approx \frac{8000(2.2251)}{0.05}$

   $S_{24}\approx \frac{17800.8}{0.05}$

   $S_{24}\approx 356016$

The total amount she will pay in 24 installments is approximately Rs. 356,016.

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Key Formulas or Methods Used

The primary formula used is the one for the Sum of a Finite Geometric Series:

$S_n=\frac{a_1(r^n-1)}{r-1}$

Where:

• $S_n$ is the sum of the first $n$ terms.
• $a_1$ is the first term of the sequence.
• $r$ is the common ratio.
• $n$ is the number of terms.

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Summary of Steps

1. Identify the sequence as geometric with a first term $a_1=8000$.
2. Calculate the common ratio based on the 5% increase: $r=1+0.05=1.05$.
3. Set the number of terms to $n=24$.
4. Apply the formula for the sum of a geometric series: $S_n=\frac{a_1(r^n-1)}{r-1}$.
5. Substitute the values and calculate the sum to find the total amount paid, which is approximately Rs. 356,016.