Applications of Arithmetic and Geometric Progressions

Returning to the chapter opening investigation about the Koch snowflake, the enclosed area can be found using the sum of an infinite series.
In the second iteration, since the sides of the new triangles are $\frac{1}{3}$ the length of the sides of the original triangle, their areas must be ${\left(\frac{1}{3}\right)}^2=\frac{1}{9}$ of its area.
If the area of the original triangle is $1$ square unit, then the total area of the three new triangles is $3\left(\frac{1}{9}\right)$.
Find the total area for the second iteration in square units.

Returning to the chapter opening investigation about the Koch snowflake, the enclosed area can be found using the sum of an infinite series.
In the second iteration, since the sides of the new triangles are $\frac{1}{3}$ the length of the sides of the original triangle, their areas must be ${\left(\frac{1}{3}\right)}^2=\frac{1}{9}$ of its area.
If the area of the original triangle is $1$ square unit, then the total area of the three new triangles is $3\left(\frac{1}{9}\right)$.
Find the total area for the third iteration in square units.

What is the number of moves required in the Tower of Hanoi problem for $k$ disks?

Write a general formula to calculate the amount remaining of the substance. If you start with a $60-\mathrm{gram}$ sample of the isotope, how much will remain  in $7.2$ years.

Half-life is the time required for a substance to decay to half of its original amount. A radioactive isotope has a half-life of $1.23$ years. Explain what this means.

Write a general formula to calculate the amount remaining of the substance. If you start with a $52-\mathrm{gram}$ sample of the isotope, how much will remain  in $7.2$ years.

Half-life is the time required for a substance to decay to half of its original amount. A radioactive isotope has a half-life of $1.23$ years. Explain what this means.

Find the minimum number of moves required to solve a Tower of Hanoi puzzle, if there are $5$ disks.

Find the minimum number of moves required to solve a Tower of Hanoi puzzle, if there are $3$ disks.

A sum of $\mathrm{Rs}.1000$ is invested at $8\%$ simple interest per year. Calculate the interest at the end of each year. Do these interests form an Arithmetic Progression? Explain.

A sum of money becomes four times at simple interest rate of $5\%$. At what rate it becomes seven times?

A certain amount of simple interest of $Rs.2500$ after $5$ years. Had the interest increased $2\%$ more, what is SI now?(Assume $P= 10,000$)

David invested certain amount in three different schemes A, B and C with the rate of interest $10\% p.a$., $12\% p.a$. and $15\% p.a$. respectively. If the total interest accrued in one year was $Rs.3200$ and the amount invested in Scheme C was $150\%$ of the amount invested in Scheme A and $240\%$ of the amount invested in Scheme B, what was the amount invested in Scheme B?

A woman invests $\mathrm{Rs}. 200$ at the start of each year at $5\%$ compound interest per annum. How much will her investment be at the end of the $2^{\mathrm{nd} }$ year?

A woman invests Rs. $2000$ at the start of each year at $5\%$ compound interest per annum. How much will her investments be at the end of the $2^{\mathrm{nd} }$ year?

Marek invested PLN $4500$ (Polish ztoty) in a bank that paid $r\%$ interest per annum compounded quarterly. After six years he had PLN $\$5179.27$ in the bank. Find the interest rate.(Use: $\ln 1.1508=0.14045$)