Free IB DP Practice Questions | RevisionPrep

1ChallengeLAQSum of an arithmetic sequences10 marksPaper 3~15 min

A manufacturing company produces cylindrical metal rods. The rods are produced in batches, and the quality control department inspects them for defects. The number of rods inspected each day forms an arithmetic sequence. On the first day,

15

rods are inspected. On each subsequent day, the number inspected increases by a constant amount

d

, where

d > 0

.

(a)

Starting from the definition of an arithmetic sequence

u_k = 15 + (k-1)d

, use the identity

\displaystyle S_n = \sum_{k=1}^{n} u_k

and the standard result

\displaystyle\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

to prove that

S_n = \frac{n}{2}\bigl(30 + (n-1)d\bigr).

[5 marks]

(b)

After

20

days, the total number of rods inspected is

2100

. Calculate the value of

d

. [2 marks]

(c)

The factory's inspection line has a maximum daily capacity of

200

rods. Determine the day on which the number of rods inspected first reaches or exceeds

200

, and hence calculate the total number of rods inspected up to and including that day. [3 marks]

Solutions

2ChallengeLAQSum of an arithmetic sequences8 marksPaper 3~12 min

A financial analyst models the depreciation of a specialised machine. The machine is purchased for USD 50000. At the end of the first year its value is USD 47000, and the value decreases by the same fixed amount each year, forming an arithmetic sequence.

(a)

Show that the total value of the machine (the sum of its values at the end of each year) over the first

n

years is given by

S_n = \frac{n}{2}(97000 - 3000n).

[4 marks]

(b)

Calculate

S_{10}

, the total value accumulated over the first 10 years. [2 marks]

(c)

The analyst claims the model is only physically valid while the machine retains a non-negative value. (i) Determine the maximum number of complete years for which the model is valid. [2]
(ii) Hence find the value of

n

that maximises

S_n

within the valid domain, and state this maximum total value. [2 marks]

Solutions

3ChallengeLAQDefinition and general term of arithmetic sequencess8 marksPaper 3~12 min

A sequence

\{u_n\}

is defined by

u_1 = a

, where

a \in \mathbb{R}^+

, and the recurrence relation

u_{n+1} = u_n + d

for

n \geq 1

, where

d

is a non-zero real constant.

(a)

Prove by mathematical induction that

u_n = a + (n-1)d

for all

n \in \mathbb{Z}^+

. [3 marks]

(b)

Let

S_n = \displaystyle\sum_{k=1}^{n} u_k

. Prove that

S_n = \dfrac{n}{2}\bigl(2a + (n-1)d\bigr)

. [3 marks]

(c)

Consider the sequence

\{v_n\}

defined by

v_n = u_{2n} - u_n

for

n \geq 1

. - (i) Prove that

\{v_n\}

is an arithmetic sequence, stating its first term and common difference. [2 marks] -
(ii) Given that

u_1 = 5

and

u_3 = 9

, find the value of

n

such that

v_n = 100

. [2 marks]

Solutions

4ChallengeLAQDefinition and general term of arithmetic sequencess9 marksPaper 3~14 min

A sequence

\{w_n\}

is defined by

w_1 = p

, where

p \in \mathbb{R}

, and

w_{n+1} = w_n + q

, where

q

is a non-zero real constant.

(a)

Prove by mathematical induction that

w_n = p + (n-1)q

for all

n \in \mathbb{Z}^+

. [3 marks]

(b)

Let

T_n = \displaystyle\sum_{k=1}^{n} w_k

. Prove that

T_n = \dfrac{n}{2}\bigl(2p + (n-1)q\bigr)

. [2 marks]

(c)

A new sequence

\{x_n\}

is defined by

x_n = w_{2n} - w_{2n-2}

for

n \geq 1

. (i) Show that

\{x_n\}

is an arithmetic sequence, stating its first term and common difference. [3 marks]
(ii) Hence find

\displaystyle\sum_{k=1}^{m} x_k

in terms of

m

,

p

, and

q

, and verify your answer is consistent with the result in (b). [4 marks]

Solutions

5ChallengeLAQDefinition and general term of arithmetic sequencess9 marksPaper 3~14 min

A sequence

\{u_n\}

is defined by

u_1 = p

, where

p

is a positive integer, and for

n \geq 1

,

u_{n+1} = u_n + k,

where

k

is a non-zero constant.

(a)

Prove that

\{u_n\}

is arithmetic, stating its common difference. [2 marks]

(b)

Derive the general term

u_n

in terms of

p

,

k

, and

n

. Consider a second sequence

\{w_n\}

defined by

w_n = u_n^2

. [2 marks]

(c)

Show that

w_{n+1} - w_n = 2pk + (2n-1)k^2.

[3 marks]

(d)

Hence prove that

\{w_n\}

is not arithmetic for any non-zero value of

k

. [1] (e) The sequence of differences

d_n = w_{n+1} - w_n

is itself arithmetic for particular values of

p

and

k

. (i) State the common difference of

\{d_n\}

in terms of

k

. [1]
(ii) For the case

p = 5

and

k = 2

, find the least value of

N

such that

d_N > 200

. [2 marks]

Solutions

6ChallengeLAQDefinition and general term of arithmetic sequencess8 marksPaper 3~12 min

A new type of sequence, called a harmonic-arithmetic sequence, is defined as follows. For given non-zero real numbers

a

and

d

, the sequence

\{u_n\}

is defined by

u_1 = a

, and for

n \geq 1

,

\frac{1}{u_{n+1}} = \frac{1}{u_n} + d.

(a)

Prove that the sequence

\{v_n\}

defined by

v_n = \dfrac{1}{u_n}

is arithmetic. State its first term and common difference in terms of

a

and

d

. [2 marks]

(b)

Hence show that the general term of the original sequence is

u_n = \frac{a}{1 + a(n-1)d},

and state any condition

n

required for

u_n

to be defined. [3 marks]

(c)

Consider the specific case

a = 2

,

d = 0.1

. (i) Calculate

\displaystyle\lim_{n \to \infty} u_n

, justifying your answer. [2]
(ii) A student claims: "For any non-zero $a$ , the limit of $u_n$ as $n \to \infty$ is always $0$ , provided $d \neq 0$ ." Analyse this claim. Your answer should consider the effect of the sign of

d

and whether any terms of the sequence may be undefined. [3 marks]

Solutions

7ChallengeLAQSum of a geometric sequences7 marksPaper 3~11 min

A financial analyst models the growth of an investment fund. The fund's value at the start of each month forms a geometric sequence. At the beginning of month 1 the fund is worth USD 1000. At the beginning of month 2 it is worth USD 1100, and at the beginning of month 3 it is worth USD 1210. Let

V_n

represent the value of the fund (in dollars) at the beginning of month

n

, for

n \in \mathbb{Z}^+

.

(a)

Show that

V_n

forms a geometric sequence, and state the values of

a

and

r

. [2 marks]

(b)

Show that the total value accumulated over the first

n

months,

T_n = V_1 + V_2 + \cdots + V_n

, satisfies

T_n = 10000\!\left(1.1^n - 1\right).

[3 marks]

(c)

The analyst claims that the fund's total accumulated value first exceeds USD 21000 during month 12. (i) Determine the smallest integer

n

such that

T_n > 21000

. [3]
(ii) Hence evaluate whether the analyst's claim is correct. Justify your answer. [2 marks]

Solutions

8ChallengeLAQSum of an arithmetic sequences7 marksPaper 3~11 min

Data

$

(ii) Hence find

S_{2k+4}

in terms of

a

and

d

, justifying your answer fully.

\quad [2]

(iii) Using the result of part (b)(i), determine the value of

n

for which

S_n

is maximum, giving your answer in terms of

k

. Hence state the maximum value of

S_n

in terms of

a

and

d

.

\quad [4]

Data summary:

u_1 = a > 0

,

d < 0

,

S_k = S_{k+5}

.

An arithmetic sequence

\{u_n\}

has first term

u_1 = a

and common difference

d

, where

a > 0

and

d < 0

.

S_n

denotes the sum of the first

n

terms.

(a)

By writing

S_n

in two different orders and adding, show that $$S_n = \frac{n}{2}\bigl(2a + (n-1)d\bigr). \quad [3 marks]

(b)

It is given that

S_k = S_{k+5}

for some positive integer

k

. (i) Show that $2a + d(2k + 4) = 0$. $\quad [4 marks]

Solutions

9ChallengeLAQSum of a geometric sequences10 marksPaper 3~15 min

A geometric sequence has general term

u_n = ar^{n-1}

for

n \in \mathbb{Z}^+

, where

a

is the first term and

r

is the common ratio,

r \neq 1

. The sum of the first

n

terms is

S_n = a + ar^2 + \cdots + ar^{n-1}

.

(a)

For the geometric sequence with

a = 1

and

r = 2

, show that

S_n = 2^n - 1

. [3 marks]

(b)

For a geometric sequence with first term

a

and common ratio

r \neq 1

, prove that

S_n = \frac{a(r^n - 1)}{r - 1}.

[4 marks]

(c)

Consider the sum

\displaystyle T = \sum_{k=0}^{m} 3 \cdot 2^k

. (i) State the values of

a

,

r

, and the number of terms in

T

, identifying

T

as a geometric series. [1]
(ii) Hence show that

T = 3(2^{m+1} - 1)

. [2]
(iii) Determine the least integer value of

m

such that

T > 3000

. [3 marks]

Solutions

10ChallengeLAQSum of a geometric sequences8 marksPaper 3~12 min

A geometric sequence has first term

u_1 = 2

and common ratio

r = 3

.

(a)

Prove that the sum of the first

n

terms is

S_n = 3^n - 1

. [3 marks]

(b)

Find the smallest value of

n

such that

S_n > 10^6

. [3 marks]

(c)

A different geometric sequence has first term

a

and common ratio

r

, where

r \neq 1

. (i) Show that the sum of the first

n

terms is

\displaystyle S_n = \frac{a(r^n - 1)}{r - 1}

. [2 marks]
(ii) Hence determine all pairs

(a,\, r)

with

a \in \mathbb{Z}^+

,

r \neq 1

, for which the sum formula simplifies to

S_n = r^n - 1

. Justify your answer. [2 marks]

Solutions

11ChallengeLAQSum of a geometric sequences10 marksPaper 3~15 min

A geometric sequence is defined by

u_1 = a

and

u_{n+1} = r\,u_n

for

n \in \mathbb{Z}^+

, where

r

is a non-zero constant. The sum of the first

n

terms is

S_n = a + ar^2 + \cdots + ar^{n-1}

.

(a)

Prove that

S_n = \dfrac{a(1 - r^n)}{1 - r}

for

r \neq 1

. [4 marks]

(b)

A geometric sequence has first term

a = 3

and common ratio

r = \dfrac{1}{2}

. Calculate

S_{10}

, giving your answer as an exact fraction. [2 marks]

(c)

The infinite sum of this sequence is

S_\infty = \dfrac{a}{1-r}

for

|r| < 1

. Show that the tail sum

S_\infty - S_n = \dfrac{a\,r^n}{1 - r}

, and hence find the least integer

N

such that

S_\infty - S_N < 0.001

. [4 marks]

Solutions

12ChallengeLAQSum of an arithmetic sequences10 marksPaper 3~15 min

A company is developing a new type of solar panel. The energy output of the first panel tested is

120\,\text{kWh}

. Each subsequent panel tested produces an energy output that is a constant amount less than the previous panel, due to manufacturing tolerances. The energy outputs of the panels form an arithmetic sequence. The total energy output of the first

20

panels tested is

1880\,\text{kWh}

.

(a)

Prove that the common difference,

d

, of the arithmetic sequence is

-4\,\text{kWh}

. [3 marks]

(b)

The company decides that any panel with an energy output of less than

50\,\text{kWh}

is considered defective and will not be sold. Determine the number of non-defective panels among the first

50

panels tested. Justify your answer. [3 marks]

(c)

The company claims that the total energy output of all non-defective panels among the first

50

panels exceeds

1500\,\text{kWh}

. Determine whether this claim is correct, showing all working. [4 marks]

Solutions

13ChallengeLAQGeometric sequences and their general terms10 marksPaper 3~15 min

A geometric sequence

\{v_n\}

has first term

b

and common ratio

q

, where

b > 0

and $0 < q < 1$. The sum to infinity exists and equals

4b

.

(a)

Prove that

q = \dfrac{3}{4}

. [3 marks]

(b)

Show that the sum of the first

m

terms satisfies

S_m = 4b\!\left(1 - \left(\frac{3}{4}\right)^{\!m}\right).

[3 marks]

(c)

Find the least integer

m

such that

S_m > \dfrac{255}{64} b

, giving your answer as an exact value obtained from an inequality involving logarithms. [4 marks]

Solutions

14ChallengeLAQGeometric sequences and their general terms10 marksPaper 3~15 min

A geometric sequence

\{u_n\}

has first term

a > 0

and common ratio

r > 1

. The sum of the first

n

terms is

S_n

.

(a)

Show that

S_n = \dfrac{a(r^n - 1)}{r - 1}

by multiplying

S_n

by

r

, subtracting, and factorising. [3 marks]

(b)

The sequence satisfies the condition that the sum of the first

k

terms equals the sum of the terms from position

k+1

to position

2k

, for some positive integer

k

. Show that this condition leads to

r^{2k} - 2r^k + 1 = 0.

[4 marks]

(c)

By treating the equation in (b) as a quadratic in

r^k

, find all solutions for

r^k

and hence determine the value(s) of

r

that satisfy the original condition. Explain, with justification, why no value

r > 1

can satisfy the condition, and determine the range of values of

r

for which a real solution

r^k > 0

could exist. [3 marks]

Solutions

Arithmetic and Geometric Sequences · Paper 3 — Free Maths AA HL Practice Questions