Geometric Progressions

Evaluate: $\sum _{n=1}^6\left(\frac{2}{5}\right){\left(3\right)}^n$

If $G$ be the geometric mean between two given numbers and $A_1, A_2$ be the arithmetic means between them. Then, $G^2$

Let $a, b, c$ be in an AP and $a^2, b^2, c^2$ be in GP. If $a<b<c$ and $a+b+c=\frac{3}{2}$, then find the value of $a$

If $y,x,z$ are in $\mathrm{A}.\mathrm{P}.$, then

If $1+r+r^2+\dots \dots \dots +r^n=\left(1+r\right)\left(1+r^2\right)\left(1+r^4\right)\left(1+r^8\right)$, then the value of $n$ is

If $x\ne 0$ and $\left(1+x+x^2\right)\left(1+x+\dots +x^{10}\right)={\left(1+x+\dots +x^6\right)}^2$ then $x+5$ is equal to

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

Let both the roots of equation $x^2+2\left(P-3\right)x+9=0,P\in N$ lie in the interval $\left(-6,1\right)$. If $20$ geometric means $g_i$ where $i=1,2,3,\dots .20$ are inserted between $2$ and $P$, then $g_6{g}_{12}$ is equal to

Let $A_1,{ A}_2,{ A}_3,\dots \dots .$ be an increasing geometric progression. If $\frac{1}{{ A}_4}+\frac{1}{{ A}_8}=\frac{10}{63}$ and $A_5{ A}_7=441$, then $3\left({ A}_2+A_4\right)$ is equal to ___________.

Let $a_1, a_2, a_3, a_4, a_5$ be a $\mathrm{GP}$ of positive real number such that the $\mathrm{AM}$ of $a_2$ and $a_4$ is $117$ and the $\mathrm{GM}$ of $a_2$ and $a_4$ is $108$. Then, the $\mathrm{AM}$ of $a_1$ and $a_5$ is

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)