Geometric Progression

If $1, 3, 3^2, 3^3, \dots .....,3^{2021}$ is divided by $50$, then remainder is

Determine whether the infinite geometric series below are converging, and also, find the sum for the following series : $0.25+0.375+0.5625+\dots$

Find the number of terms in the following geometric progression.

$4,12,36,...,8748$

Find the number of terms in the following geometric progression.

$4,12,36,...,78732$

Find the number of terms in the following geometric progression.

$3,6,12,24,....,384$

Find the number of terms in the following geometric progression.

$2,4,8,\dots ,131072$

Find the number of terms in the following geometric progression.

$2,4,8,\dots ,1024$

At the end of $2016$ the population of a city was $200000$. At the end of $2018$ the population was $264500$. Comment on whether this increase will continue. Give a reason for your answer.

Carlos decides to take out a loan of $20000$ Peruvian Soles (SOL) to buy a car. His bank offers two options to finance the loan. 

Option A: A five-year loan with an annual interest rate of $12.8\%$ compounded quarterly. No deposit required.

Option B: A five-year loan with an annual interest rate of $r\%$ compounded monthly. Terms of the loan require a $10\%$ deposit and monthly repayments of SOL $400$.

If Carlos chooses option B, find the annual interest rate, $r$. (Use: ${\left(\frac{4}{3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$60$}\right.}=1.0048$)

Carlos decides to take out a loan of $20000$ Peruvian Soles (SOL) to buy a car. His bank offers two options to finance the loan. 

Option A: A five-year loan with an annual interest rate of $12.8\%$ compounded quarterly. No deposit required.

Option B: A five-year loan with an annual interest rate of $r\%$ compounded monthly. Terms of the loan require a $10\%$ deposit and monthly repayments of SOL $400$.

State which option Carlos should choose. Justify your reasoning.

Katharina and carolina go to a swimming pool. they both swim the first length of the pool in $2$ minutes $6$ seconds.

The time that katharina takes to swim each subsequent length is $5$ seconds more than she look to swim the previous length.

The time that carolina takes to swim each subsequent length is $1.04$ times that which she took to swim the previous length.

Show that carolina takes $2$ minutes and $16$ seconds (to the nearest second) to swim the third length.

Mr Garcia is offered a $10$ -year contract with a starting annual salary of $20$ million Colombian pesos (COP) and yearly increases of $5\%$ on the previous annual salary. After considering the company offer, Mr Garcia asks for a different payment plan. He wants to start with the same annual salary, but have fixed annual pay rises of $x$ thousand pesos (COP). The company financial manager agrees to this new payment scheme if the total amount for the $10$ years remains the same as it was the original payment offer. Determine the value of $x$, correct to the nearest peso, that satisfies the financial manager's conditions. Hence, determine the number of years for which the second plan gives Mr Garcia a higher income than the first.

Mr Garcia is offered a $10$ -year contract with a starting annual salary of $20$ million Colombian pesos (COP) and yearly increases of $5\%$ on the previous annual salary. After considering the company offer, Mr Garcia asks for a different payment plan. He wants to start with the same annual salary, but have fixed annual pay rises of $x$ thousand pesos (COP). The company financial manager agrees to this new payment scheme if the total amount for the $10$ years remains the same as it was the original payment offer. Determine the value of $x$, correct to the nearest peso, that satisfies the financial manager's conditions.

Mr Garcia is offered a $10$ -year contract with a starting annual salary of $20$ million Colombian pesos (COP) and yearly increases of $5\%$ on the previous annual salary. Find the total amount Mr Garcia will earn during his 10 years at the company, correct to the nearest thousand pesos.

Mr Garcia is offered a $10$ -year contract with a starting annual salary of $20$ million Colombian pesos (COP) and yearly increases of $5\%$ on the previous annual salary. The value of Mr Garcia's annual salary during the third year of his contract is equal to $k$ million COP. Find the value of $k$.

Mr Garcia is offered a $10$ -year contract with a starting annual salary of $20$ million Colombian pesos (COP) and yearly increases of $5\%$ on the previous annual salary. Justify that Mr Garcia's annual salaries over the $10$ -year period form a geometric sequence, and state its common ratio. 

Giovanna works on a farm and feeds the chickens each week. At the beginning the farm has $500$ chickens. The number of chickens increases each week by $1\%$. Find the total number of feeds in $15$ weeks. (Give the answer in integer value)

Giovanna works on a farm and feeds the chickens each week. At the beginning the farm has $500$ chickens. The number of chickens increases each week by $1\%$. Find the number of chickens after $15$ weeks.

Find the sum of this geometric series: $1+3+9+\dots +19683$

Alexa wins a prize in a lottery. She receives CAD $8000$ (Canadian dollars) the first month, CAD $6000$ the second month, CAD $4500$ the third month and so on for a total of six months. Calculate how much she receives in month six. (Round off the answer to two decimal place)