Question Statement

You save Rs. 1 on the first day. On each subsequent day, you save double the amount you saved the day before. Find the amount you should save on the 20th day of your plan.

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Background and Explanation

This problem involves a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). In this scenario, the daily savings amount grows exponentially, making it a perfect example of a geometric progression.

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Solution

To find the savings on the 20th day, we first need to define the geometric sequence based on the problem statement.

1. Identify the Sequence:

   • Saving on the $1^{st}$ day: $a_1=1$
   • Saving on the $2^{nd}$ day: $a_2=1\times 2=2$ (double the 1st day)
   • Saving on the $3^{rd}$ day: $a_3=2\times 2=4$ (double the 2nd day) The sequence of savings is 1, 2, 4, ...

2. Determine the First Term and Common Ratio:

   • The first term ($a_1$) is the amount saved on day 1, which is 1.
   • The common ratio ($r$) is the factor by which the savings increase each day. Since the amount doubles, $r=\frac{2}{1}=\frac{4}{2}=\text{2}$.

3. Find the Savings on the 20th Day ($a_{20}$): We use the formula for the $n^{th}$ term of a geometric sequence. We need to find the value of the 20th term, so $n=20$.

   The formula is: $a_n=a_1\cdot r^{n-1}$

   Substitute the known values ($a_1=1$, $r=2$, and $n=20$) into the formula:

   $a_{20}=1\cdot (2{)}^{20-1}$

   $a_{20}=1\cdot (2{)}^{19}$

   $a_{20}=524,288$

Therefore, the amount you should save on the 20th day is Rs. 524,288.

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Key Formulas or Methods Used

The key formula used to solve this problem is the formula for the $n^{th}$ term of a geometric sequence:

$a_n=a_1\cdot r^{n-1}$

Where:

• $a_n$ is the term we want to find (the savings on the $n^{th}$ day).
• $a_1$ is the first term in the sequence (savings on day 1).
• $r$ is the common ratio.
• $n$ is the term number (the specific day).

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Summary of Steps

1. Identify the pattern of savings (1, 2, 4, ...) as a geometric sequence.
2. Determine the first term ($a_1=1$) and the common ratio ($r=2$).
3. Set the term number to $n=20$, as we need the savings on the 20th day.
4. Apply the $n^{th}$ term formula: $a_n=a_1\cdot r^{n-1}$.
5. Substitute and calculate: $a_{20}=1\cdot (2{)}^{19}=524,288$.