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Analysis and Approaches SL — Questionbank

All Concepts - Number & Algebra

All questions in Topic 1 Number & Algebra

Question 1

NO CALCULATOR

Mild

[Maximum mark: 6]

Nickel in the asteroid 433 Eros is estimated to be worth 9457 quadrillion euros (EUR).
(Note: 1 quadrillion =10¹⁵)

(a) Write this value in standard scientific notation, in the form. a × 10^k, where 1 ≤ a < 10, k ∈ Z.

Ashley assumes that the asteroid is approximately spherical with a radius of 146 km.

[2]

(b) Using this assumption, calculate Ashley’s estimate of the asteroid’s volume, in km³

The actual volume of the asteroid is known to be. 9.72 x 10⁶

[2]

(c) Calculate the percentage error in James’s estimate compared to the actual volume.

[2]

Question 2

NO CALCULATOR

Mild

[Maximum mark: 6]

The first three terms of an arithmetic sequence are

${u_{1},\ 4u - 9,\ 2u}_{1} + 27$

(a) Show that u₁ = 9

Ashley assumes that the asteroid is approximately spherical with a radius of 146 km.

[2]

(b) Prove that the sum of the first n terms of this arithmetic sequence is always a square number.

[4]

Question 3

NO CALCULATOR

Mild

[Maximum mark: 4]

A sphere with diameter 12 104 000 metres can model the shape of the Venus.


(a) Use this model to calculate the circumference of Venus in kilometres. Round your answer to three significant figures	[3]
(b) Write your answer from (a) in the form a × 10^k, where 1 ≤ a < 10 , k ∈ Z.	[1]

Question 4

NO CALCULATOR

Medium

[Maximum mark: 6]

For a particular arithmetic sequence,

u_{12} = 20\ and\ S_{18} = 342

Find the value of k such that

u_{k} = 2

Question 5

NO CALCULATOR

Spicy

[Maximum mark: 16]

Consider the arithmetic sequence

b, m, n,\ldots

where $b, m, n \neq 0$


(a) Show that 2m − n = b Now consider the geometric sequence b, r, u, … where b, r, u ≠ 0	[2]
(b) Show that r² = bu The first term of both sequences is b. It is given that n = u = 9	[1]
(c) Show that m > 3 Consider the case where b = 25, r > 0 and n = u = 9	[2]
(d.i) Write down the first four terms of the arithmetic sequence	[2]
(d.ii) Write down the first four terms of the geometric sequence A new arithmetic sequence (v_k)is formed by adding together corresponding terms of the two sequences: v_k = (term from arithmetic sequence) + ln (term from geometric sequence). It is given that v₁ = 25 + ln 25, v₂ = 17 + ln 15,	[2]
(e.i) Find the common difference of the sequence (v_k) in terms of ln 3 and ln 5	[2]
(e.ii) Show that $\sum_{k = 1}^{10}v_{k} = - 110 - 25\ln 5 + 45\ln 3$	[4]

Question 6

NO CALCULATOR

Medium

[Maximum mark: 7]

The sum of the first n terms of an arithmetic sequence is given by

S_{n} = \ \frac{1}{2}\left( an^{2} + bn \right)

where a and b are constants.

It is given that $S_{3} = 18\ and\ S_{5} = 50$


(a) Find the value of a and the value of b.	[5]
(b) Find the value of $u_{5}$	[2]

Question 7

NO CALCULATOR

Spicy

[Maximum mark: 14]

Consider the arithmetic sequence

u_{1,\ \ }\ u_{2},\ u_{3}\ldots

The sum of the first n terms of this sequence is given by

S_{n} = \ n^{2} + 6n


(a.i) Find the sum of the first four terms	[2]
(a.ii) Given that S₇ = 91, find the value of u₇	[2]
(b) Find the value of u₁	[2]
(c) Write the expression for u_n Now consider a geometric sequence v_n, where v₃ = u₂ and v₅ = u₈	[3]
(d) Find the possible values of the common ratio r	[3]
(e) Given that v₅₀ < 0, determine the value of v₆	[2]

Question 8

NO CALCULATOR

Spicy

[Maximum mark: 10]

Consider an arithmetic sequence with

$u_{1} = 0.5\ \ and\ u_{4} = 0.2$

(a) Find the common difference, d.

The following table shows the probability distribution of a discrete random variable X, where

P(X = n) = u_nk, n ∈ Z⁺, 1 ≤ n ≤ 4, k ∈ R⁺

n	1	2	3	4
P(X = n)	0.5k	u₂k	u₃k	0.2k

[2]

(b) Find the value of k

[3]

(c) Hence, write down the probability distribution of X.

[2]

(d) Find the expected value E(X)

[3]

Question 9

NO CALCULATOR

Spicy

[Maximum mark: 12]

Consider the series

\ln x + q\, lnx + \frac{1}{4}\ln x + \ldots,

where $x \in R, x > 1, q \in R, q \neq 0$


(a.i) Show that if the series is geometric then $q = \ \pm \frac{1}{2}$	[2]
(a.ii) Given that q > 0, and S_∞ = 4 + 4, find the value of x Now consider the case where the series is arithmetic with common difference d.	[3]
(b.i) Show that in this case $q = \ \frac{5}{8}$	[3]
(b.ii) Write down d in the form mln x, where m ∈ Q	[1]
(b.iii) Suppose the sum of the first n terms of the series equals $- \frac{7}{8}\ln x$ . Find the value of n.	[3]

Question 10

NO CALCULATOR

Mild

[Maximum mark: 4]

The nth term of an arithmetic sequence is given by

u_{n} = 20 - 4n.


(a) State the value of the first term, $u_{1}$	[1]
(b) Given that the n^th term of this sequence is −28, find the value of n.	[2]
(c) Find the common difference, d	[2]

Question 11

NO CALCULATOR

Mild

[Maximum mark: 4]

Consider an arithmetic sequence where

u_{6} = S_{6} = 12

Find the value of the first term, $u_{1}$ and the value of the common difference, $d$

Question 12

CALCULATOR

Mild

[Maximum mark: 5]

A geometric sequence has a first term of 20 and a fourth term of 67.5

The sum of the first n terms of the sequence is denoted by $S_{n}$ .

Find the smallest value of n such that

S_{n} > 10\, 000

Question 13

CALCULATOR

Medium

[Maximum mark: 6]

An arithmetic sequence has first term 5 and common difference −2.

(a) Given that the k^th term of the sequence is one, find the value of k.

[2]

(b) Let S_n denote the sum of the first n terms of the sequence.

Find the maximum value of S_n

[4]

Question 14

CALCULATOR

Medium

[Maximum mark: 7]

The first terms of an infinite geometric sequence, $u_{n}$ , are 3, 9, 27, 81, …

The first terms of a second infinite geometric sequence, $v_{n}$ , are 3, - 9, 27, - 81, …

The terms of a third sequence, $w_{n}$ , are defined as $w_{n} = u_{n} + v_{n}.$

The finite series, $\sum_{k = 1}^{150}w_{k}$ can also be written in the form $\sum_{k = 0}^{m}{6r^{k}}$


(a) Write down the first three non-zero terms of $w_{n}$	[3]
(b.i) Find the value of $r$	[2]
(b.ii) Find the value of $m$	[2]

Question 15

CALCULATOR

Spicy

[Maximum mark: 12]

In an arithmetic sequence, $u_{1} = 2.5,\ \ u_{2} = 2.8\ and\ u_{k} = 50.2.$


(a) Find the value of k for which u_k = 50.2	[2]
(b) Find the exact value of S_k Consider the terms, u_n, of this sequence such that n ≤ k. Let F be the sum of the terms for which n is not a multiple of 4.	[2]
(c) Show that F = 3144	[5]
(d) An infinite geometric series is given as $S_{\infty} = a + \frac{a}{2} + \frac{a}{4} + \ldots,\ a \in Z^{+}.$ Find the largest value of a such that S_∞< F.	[3]

Question 16

CALCULATOR

Medium

[Maximum mark: 11]

Maria purchases a new laptop for 1200 US dollars (USD) and pays with a Canadian credit card. There is a transaction fee of 3.8% charged by the credit card company to convert the purchase into Canadian dollars (CAD).

The exchange rate is 1 USD = 1.40 CAD.


(a) Calculate, in CAD, the total amount Maria pays for the laptop. The laptop’s value depreciates each year. The values of the laptop form a geometric sequence. At the end of the first year the value is 900 USD, and at the end of the second year the value is 720 USD.	[3]
(b) Find the value of the laptop at the end of the 5th year. Give your answer to two decimal places.	[3]
(c) Calculate, in years, when the value of the laptop will first fall below 100 USD. Maria insures her laptop with a US company. During the 1st year she pays 150 USD to insure the laptop. Each year the insurance premium decreases by 4.50 USD.	[2]
(d) Find the total amount Maria pays to insure the laptop for the first 6 years.	[3]

Question 17

CALCULATOR

Medium

[Maximum mark: 12]

Alan and Maria both commence new jobs each starting on an annual salary of $60000

At the start of each new year, Alan receives an annual salary increase of $2000

Let $ $A_{n}$ represent Alan’s annual salary at the start of his $n^{th}$ year of employment.


(a) Show that A_n = 2000n + 58000 At the start of each new year, Maria receives an annual salary increase of 2.5% of her previous year’s annual salary. Maria’s annual salary, M_n, at the start of her n^th year of employment is given by M_n = 60000(1.025)^n − 1	[2]
(b) Given that M_n follows a geometric sequence, state the value of the common ratio, r At the start of year N, Maria’s annual salary exceeds Alan’s annual salary for the first time.	[1]
(c.i) Find the value of N	[3]
(c.ii) For the value of N found in part (c.i), state Alan’s annual salary and Maria’s annual salary, correct to the nearest dollar.	[2]
(d) Find Maria’s total earnings at the start of her 12th year of employment. Give your answer correct to the nearest dollar.	[4]

Question 18

CALCULATOR

Medium

[Maximum mark: 16]

A large underground tank is constructed at Pearson Airport to store fuel. The tank is in the shape of an isosceles trapezoidal prism, ABCDEFGH.

AB=70 m , AF=200 m, AD=40 m, BC=40 m and CD=110 m. Angle ADC=60° and angle BCD=60°. The tank is illustrated below.

(a) Find h, the height of the tank.

[2]

(b) Show that the volume of the tank is 624 000 m³, correct to three significant figures.

Once construction was complete, a fuel pump was used to pump fuel into the empty tank. The amount of fuel pumped into the tank by this pump each hour decreases as an arithmetic sequence with terms u₁, u₂, u₃, …, u_n

Part of this sequence is shown in the table.

Hour(n)	1^st	2^nd	3^rd	…
Amount of fuel pumped into the tank in each hour, in m³ (u_n)	45000	43200	41400

[3]

(c) Write down the common difference, d.

[1]

(d) Find the amount of fuel pumped into the tank in the 13th hour.

[2]

(e.i) Find the value of n such that u_n = 0

[2]

(e.ii) Write down the number of hours that the pump was pumping fuel into the tank.

At the end of the 2nd hour, the total volume of fuel in the tank was 88 200 m³.

[1]

(f) Find the total amount of fuel pumped into the tank in the first 8 hours.

[2]

(g) Show that the tank will never be completely filled using this pump.

[3]

Question 19

NO CALCULATOR

Mild

[Maximum mark: 5]

Consider a geometric sequence with first term 2 and common ratio 5.

Let $S_{n}$ be the sum of the first n terms of the sequence.

(a) Find an expression S_n for in the form

$\frac{a(5^{n} - 1)}{b}\ where\ a,b \in Z^{+}$

[1]

(b) Hence, show that

S_{1} + S_{2} + S_{3} + \ldots + S_{n} = \frac{5^{n + 1} - 5 - 4n}{8}

[4]

Bottom of Form

Question 20

NO CALCULATOR

Medium

[Maximum mark: 14]

Consider the arithmetic sequence $b,\ m,\ n,\ldots$ where $b,m,n \neq 0$


(a) Show that 2m − n = b Consider the geometric sequence b, u, v… where b, u, v ≠ 0	[2]
(b) Show that u² = bv The first term of both sequences is b. It is given that n = v = 4	[2]
(c) Show that m > 2 Consider the case where b = 16, u > 0, and n = v = 4	[2]
(d.i) Write down the first four terms of the arithmetic sequence	[2]
(d.ii) Write down the first four terms of the geometric sequence A new arithmetic sequence w_n is defined, where each term is obtained by adding together the corresponding term of the arithmetic sequence and the natural logarithm of the corresponding term of the geometric sequence. The first three terms are w₁ = b + ln b, w₂ = m + ln u, w₃ = n + ln v	[2]
(e.i) Find the common difference of w_n in terms of ln 2	[3]
(e.ii) Show that $\sum_{i = 1}^{10}{w_{i} = - 110 - 5\ln 2}$	[3]