Sequences & Series (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Frankie opens a savings account and deposits $400 at the start of the first year. Compound interest is paid into the account at the end of each year at an annual rate of 3%.

Show that the amount of money in the account at the end of the first year is $412.

At the start of the second year, and of each subsequent year, Frankie deposits a further $400 into the account.

Write down the interest earned during the second year by the $400 deposited at the start of the second year.

Explain why the amount of money in the account, in dollars, at the end of the second year is given by $(400\times 1.03)\times 1.03+400\times 1.03$.

Hence show that the amount of money in the account, in dollars, at the end of $n$ years is given by $400\left(1.03+1.{03}^2+1.{03}^3+\dots +1.{03}^n\right)$.

Show that the sum of the geometric progression $1.03+1.{03}^2+1.{03}^3+\dots +1.{03}^n$ is $\frac{103}{3}\left(1.{03}^n-1\right)$.

Hence find the amount of money in the account at the end of 12 years, giving your answer correct to the nearest cent.

The first and second terms of a progression are $5x$ and $x^2$, where $x$ is a constant.

For the case where the progression is arithmetic with a common difference of $14$,

(i) show that $x^2-5x-14=0$;

(ii) find the possible values of $x$ and the corresponding values of the third term of the progression.

For the case where the progression is geometric with a sum to infinity of $10$,

(i) explain why $x$ cannot be equal to $0$;

(ii) show that $\frac{5x}{1-\frac{x}{5}}=10$;

(iii) find the third term of the progression.

Two sequences model the value of a car in successive years, where $u_n$ is its value in year $n$ and year 1 is the year in which it is new. In year 1 the car's value is $25 000. Model 1 is an arithmetic progression and Model 2 is a geometric progression. In both models the value in year 8 is $7500.

Find the common difference for Model 1 and the common ratio for Model 2, giving your answers to three significant figures where appropriate.

Find the value of the car according to Model 2 in the year Model 1 predicts its value to be $5000.

State one benefit of Model 2 over Model 1 for estimating the value of older cars.

The first term of a progression is $-7x$ and the second term is $x^2$.

For the case where the progression is arithmetic with a common difference of $-6$, find the possible values of $x$ and the corresponding values of the fifth term of the progression.

For the case where the progression is geometric with a sum to infinity of $-14$, find the third term of the progression.

A model maker is constructing part of a model building using a series of hollow tubes, stacked next to each other as illustrated below.

Each tube is one-tenth shorter than the one to its left. All the tubes are the same width, as they are all cut from one longer tube.

Show that, no matter how many tubes the model maker uses, the longer tube they are cut from need not be any longer than ten times the height of the tallest tube.

In a different part of the model building, tubes are required to be stacked on top of each other, as shown below.

The tallest tube is the same length as the tallest tube in part (a). Each of the other tubes is $1$ cm shorter than the tube immediately below it. The longer tube that all the tubes are cut from is ten times the height of the tallest tube in the stack, and all of it is used.

Given that the length of the tallest tube is $a$ cm and there are $n$ tubes in the stack, show that $n^2-(2a+1)n+20a=0$.

Alex is training for a marathon. In the first week Alex runs $a$ miles, and in each subsequent week runs $d$ more miles than in the previous week. Alex trains for $n$ weeks.

Given that Alex runs $73$ miles in the last week and $702$ miles in total, show that $n(a+73)=1404$.

Given that in the week halfway through the training (week $\frac{n}{2}$) Alex runs $37$ miles, show that $nd=72$.

Find the values of $a$, $d$ and $n$.

A ball is dropped from a height of $A$ metres above a flat horizontal floor. After its first impact it bounces to a height of $x$ metres, and each subsequent bounce height is $25\%$ shorter than the previous bounce.

Show that, no matter how many times the ball bounces, the total vertical distance travelled by the ball will not exceed $(A+8x)$ metres.

State one assumption that has been made in using this model.

The first term of a progression is $x^2$ and the second term is $\frac{6x}{7}$.

For the case where the progression is arithmetic with a common difference of $-\frac{1}{7}$, find the possible values of $x$ and the corresponding values of the eighth term of the progression.

For the case where the progression is geometric with a sum to infinity of $7$, and given that $x=1$ is one possible value of $x$,

(i) find all three possible values of $x$;

(ii) determine the first term and common ratio of each corresponding progression;

(iii) show that all three progressions have the same third term, and determine that term.

A training track for cyclists is in the shape of a rectangle with a semicircle at each end, and is made up of several lanes.

The shortest, inner lane has straight runs of $l$ m, with the semicircles at each end having a diameter of $d$ m. Each lane moving outwards increases the diameter by $e$ m compared to the previous lane.

Show that the distances of each lane form an arithmetic progression with first term $(\pi d+2l)$ m and common difference $\pi e$ m.

There is a total of 12 lanes on the training track. Given that $l$, $d$ and $e$ are integers and that the total distance for all 12 laps is $96(4\pi +5)$ m, find the value of $l$ and show that $2d+11e=64$.

It is recommended that lanes are at least $2$ m wide to allow sufficient space between cyclists in different lanes. Find the least value of $e$ and the associated value of $d$.

A student is investigating a sequence of values output by a computer program reported to contain a bug. The sequence of values is $3,1,6,\frac{1}{2},9,\frac{1}{4},12,\frac{1}{8},15,\frac{1}{16},\dots$

The bug appears to have made the program output two different sequences interleaved in the same list. Suggest what the two sequences could be.

To fix the bug, the student separates the output into the odd-numbered terms and the even-numbered terms.

(i) Show that the odd-numbered terms form an arithmetic progression, and determine its first term and common difference.

(ii) Find an expression for the sum of the first $m$ terms of this progression.

Find an expression for the sum of the first $n$ even-numbered terms of the sequence of values.

The computer program also outputs the sum of the first 20 terms of the sequence of values. Find the value the computer would output, giving your answer correct to 4 decimal places.

Does the sum to infinity exist for the sequence of values output by the computer program? Give a reason for your answer.

The first three terms of an arithmetic progression are $x$, $y$ and $z$ (in that order). The first three terms of a geometric progression are also $x$, $y$ and $z$ (in that order), where $x$, $y$ and $z$ are non-zero constants.

Determine the mathematical relationship that must exist between $x$ and $y$ for this to be true, and fully describe the two progressions that result.