Arithmetic & geometric progressions

The geometric series is given by $\dfrac{3x-2}{4x}+\dfrac{(3x-2)^2}{4x}+\dfrac{(3x-2)^3}{4x}+\cdots$ where $x\ne0$. Explain why the sum to infinity exists for $\dfrac13<x<1$, and find its value in terms of $x$.

The difference between the sum of the first $n$ terms of the geometric progression and its sum to infinity is less than 0.4% of the sum to infinity. Assuming the value of $r$ is $\dfrac34$, find the least value of $n$.

An arithmetic series has first term $a$ and common difference $d$, where $a>0$ and $d$ is non-zero. The first, eighth and thirteenth terms of the arithmetic series are equal to the first three terms of a geometric series respectively.

(a)(i) The sum of the first $n$ terms of a sequence $u_1,u_2,u_3,\ldots$ is $S_n=e^2n^2-3n$. Show $u_n=e^2(2n-1)-3$.

The fifth, tenth and twentieth terms of a convergent geometric progression, G, are the first three consecutive terms of an arithmetic progression, A.

(a) In a convergent geometric progression, the sum of the first $n$ terms equals the sum of the remaining terms. Given also that the $(n+1)$th term is $\dfrac12$, determine the $(2n+1)$th term.

An arithmetic series has first term $a$ and common difference $d$, where $a,d$ are non-zero. A convergent geometric series has first term $b$ and common ratio $r$, where $b$ is positive and $r$ is non-zero. The fourth and ninth terms of the arithmetic series equal the sixth and ninth terms of the geometric series respectively, and the eleventh term of the…

A geometric series G is given by $e-e^{1+\alpha}+e^{1+2\alpha}-e^{1+3\alpha}+\cdots$, where $\alpha$ is a negative real number.

The sum of the first $n$ terms, $S_n$, is $S_n=\dfrac1a[1-(a-1)^n]$, where $a$ is a constant, $a\ne1$, $n\in\mathbb Z^+$. Obtain the $n$th term $T_n$ and prove that “$S_n$ is a geometric series” (printed wording).

(a) The first 2 terms of a GP are $p,q$, where $q<p,p\ne0$. If the sum of the first $n$ terms equals the sum to infinity of the remaining terms, prove $p^n=2q^n$.

(i) A certain machine A pumps liquid into an empty container of total volume $850\mathrm{m}^3$. The first pumping action fills it with $4\mathrm{m}^3$. Each subsequent action fills it with $0.5\mathrm{m}^3$ more than the previous action. Pumping continues until it is completely filled. Find the overflow volume at the final pumping action.

Amy cuts pieces from a long roll of ribbon. The first is 160cm and each successive piece is 8cm shorter than the preceding one.

A marathon runner and a sprinter run laps on a 400m track. The sprinter takes 56 seconds for lap 1, then 6 seconds more each subsequent lap. The marathon runner takes 1 minute for lap 1, then 4% more than the previous lap.

(a) Find the sum of all integers in $\{1,2,3,\ldots,2014\}$ that are not multiples of 3.

A hamster farm has 500 hamsters to sell. The farmer sells $k$ at the end of each week, where $k$ is a constant factor of 500 (e.g. 5,100,etc.). Price is 10 dollars in week 1 and drops by 5% each subsequent week. There is neither birth nor death.

(a) An AP has first term $a>0$ and difference $d\ne0$. Its eighth, third and second terms are the first three terms of an infinite GP.

Chloe plans to invest $k$ dollars on day 1 of each month from 1 January 2021. The bank offers:

Selena borrows 200000 dollars from a bank for a home purchase on 1 January 2025. On day 1 of each month, 0.5% interest is added, first on 1 January 2025. On each last day she pays $x$ dollars, starting 31 January 2025.

A fisherman purchases a boat for 1000 dollars at the beginning of 2011. Annual operating cost is 120 dollars in 2011 and increases by 40 dollars every year.

Besto’s coach designs a training programme starting with a 400m run on session 1. Each subsequent session covers 250m more than the previous.