No questions found
TWO-STEP SEQUENCES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at two-step sequences.
These are sequences which use two steps to get from one term to the next.
The first term in every sequence is 1.
The two steps are:
• multiply by a given number
• then add a given number.
In this question the two steps are:
• multiply by 2
• then add 1.
1st term = 1
2nd term = 1st term \times 2 + 1 = 1 \times 2 + 1 = 3
3rd term = 2nd term \times 2 + 1 = 3 \times 2 + 1 = 7
4th term = 3rd term \times 2 + 1 = 7 \times 2 + 1 = 15
(a) Work out the 5th term of this sequence.
1, 3, 7, 15, .......... [2]
(b) The $n^{th}$ term of another sequence is $2^n$.
Calculate the 2nd, 3rd and 4th terms of this sequence.
2, .......... , .......... , .......... , 32 [1]
(c) Look at your answers to part (a) and part (b).
Write down an expression, in terms of $n$, for the $n^{th}$ term of the sequence in part (a).
......................................................... [1]
506
554
528
TWO-STEP SEQUENCES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at two-step sequences.
These are sequences which use two steps to get from one term to the next.
[Image_1: The first term in every sequence is 1. The two steps are: • multiply by a given number • then add a given number.]
In this question the two steps are:
• multiply by 2
• then add 3.
The first term is 1.
(a) Work out the 2nd, 3rd and 4th terms of this sequence.
1 , .......... , .......... , .......... , 61 [2]
(b) The nth term of this sequence is $a \times 2^n + b$.
(i) Substituting $n = 1$, to get the first term of the sequence, gives the equation $2a + b = 1$.
Substitute another value for $n$ to make another equation in terms of $a$ and $b$.
................................................ [1]
(ii) Solve the simultaneous equations in part (i) to show that the nth term of the sequence is $2\times 2^n - 3$.
................................................ [2]
565
585
568
TWO-STEP SEQUENCES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at two-step sequences.
These are sequences which use two steps to get from one term to the next.
[Image_1: The first term in every sequence is 1. The two steps are: • multiply by a given number • then add a given number.]
In this question, the two steps are:
• multiply by 2
• then add 5.
The first term is 1. The expression for the $n^{th}$ term is $3 \times 2^n - 5$.
Show that this expression gives the correct value for the 4th term of this sequence. [3]
580
541
592
TWO-STEP SEQUENCES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at two-step sequences. These are sequences which use two steps to get from one term to the next.
The first term in every sequence is 1.
The two steps are:
• multiply by a given number
• then add a given number.
In this question the two steps are always:
• multiply by 2
• then add $k$.
The first term is 1.
(a) Complete the table.
Use your answer to Question 1(c) and any patterns you notice.
[Table_1]
Steps to get the next term Expression for the $n$th term
Multiply by 2, then add 1 $\text{..............................}$
Multiply by 2, then add 3 $2 \times 2^n - 3$
Multiply by 2, then add 5 $3 \times 2^n - 5$
Multiply by 2, then add 7 $\text{..............................}$
Multiply by 2, then add ......... $\text{..............................} - 9$ [2]
(b) An expression for the $n$th term of this sequence is $a \times 2^n + b$.
Find expressions for $a$ and $b$ in terms of $k$.
Write down the expression for the $n$th term of the sequence.
$a = \text{..............................}$
$b = \text{..............................}$
$n$th term $= \text{..............................}$ [3]
(c) The 5th term of a sequence using the $n$th term in part (b) is 286.
Complete the two steps.
• multiply by 2
• then add $\text{..............................}$ [3]
503
598
557
TWO-STEP SEQUENCES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at two-step sequences.
These are sequences which use two steps to get from one term to the next.
[Image_1: The first term in every sequence is 1.
The two steps are:
• multiply by a given number
• then add a given number.]
In this question the two steps are:
• multiply by 3
• then add 2.
The expression for the nth term is $a \times 3^{(n-1)} + b$.
(a) The first term is 1.
(i) Find the value of the second term of the sequence.
.................................................. [1]
(ii) Use the first two terms to write two equations in terms of $a$ and $b$.
..................................................
.................................................. [2]
(b) Find the value of $a$ and the value of $b$.
$a = ...................................................$
$b = ...................................................$ [3]
575
579
592
TWO-STEP SEQUENCES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at two-step sequences.
These are sequences which use two steps to get from one term to the next.
(a) Complete the table.
Use your answer to Question 5(b) and any patterns you notice.
[Table_1]
Steps to get the next term | Expression for the $n^\text{th}$ term
Multiply by 2, then add 1 | $2 \times 2^{(n-1)} - 1$
Multiply by 3, then add 2 |
Multiply by 4, then add 3 |
Multiply by 5, then add 4 |
Multiply by 6, then add 5 | $2 \times 6^{(n-1)} - 1$
[1]
(b) For the sequence in the last row of the table, the first term has the value 1 and the second term has the value 11.
Find which term has its value closest to 20,000,000.
[3]
501
513
588
DRIVING TO MY PLACE OF WORK (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at a model for the time that I take to drive from my home to my place of work.
I live 20 km from my place of work.
When I leave my home at 7.00 am, I drive at an average speed of 50 km/h.
(a) Calculate the time, in minutes, to drive to work when I leave home at 7.00 am.
.............................................. [3]
(b) The time that it takes me to drive to work is $m$ minutes.
Find, in its simplest form, a model for $m$ when my average speed is $v$ km/h.
.............................................. [1]
586
531
561
DRIVING TO MY PLACE OF WORK (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at a model for the time that I take to drive from my home to my place of work.
I live 20 km from my place of work. When I leave my home at 7.00 am, I drive at an average speed of 50 km/h.
When I leave home after 7.00 am, there is more traffic, and my average speed is less than 50 km/h.
My average speed decreases steadily by 1 km/h for every 2 minutes after 7.00 am that I leave home.
For example, when I leave at 6 minutes after 7.00 am, my average speed is 3 km/h less, which is 47 km/h.
(a) I leave home at 7.40 am.
(i) Find my average speed. .................................................... [2]
(ii) Show that the time to drive to work is 40 minutes. [1]
(b) I leave home x minutes after 7.00 am.
Show that a model for the time, T minutes, to drive to work is $T = \frac{2400}{100 - x}$. [2]
(c) Sketch the graph of the model $T = \frac{2400}{100 - x}$ for $0 \leq x \leq 90$. [2]
(d) I do not want to drive for more than 30 minutes.
Find the latest time that I should leave home. ................................................. [2]
(e) I must be at work by 9.00 am.
One day I oversleep and leave home at 8.35 am.
(i) Use the model to find how late I will be for work. Give your answer in hours and minutes. ............................................... [3]
(ii) Make a statement about the suitability of the model. ............................................... [1]
538
571
587
DRIVING TO MY PLACE OF WORK (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at a model for the time that I take to drive from my home to my place of work.
I live 20 km from my place of work.
When I leave my home at 7.00 am, I drive at an average speed of 50 km/h.
I leave home x minutes after 7.00 am.
(a) Explain why a model for $A$, the number of minutes after 7.00 am when I arrive at work, is
$$A = x + \frac{2400}{100 - x}.$$
............................................................................................................ [1]
(b) I must be at work by 9.00 am, which is two hours after 7.00 am.
So my maximum value of $A$ is 120.
(i) Show that, for this maximum value of $A$, $x$ is a solution to the equation
$$x^2 - 220x + 9600 = 0.$$
................................ [3]
(ii) Find this value of $x$.
..................................................... [3]
(iii) Find the latest time that I can leave home to arrive at work on time.
..................................................... [1]
573
548
538
DRIVING TO MY PLACE OF WORK (30 marks)
You are advised to spend no more than 50 minutes on this part.
This task looks at a model for the time that I take to drive from my home to my place of work.
I live 20 km from my place of work.
When I leave my home at 7.00 am, I drive at an average speed of 50 km/h.
I move to a new home and now live $d$ km from my work.
When I leave my new home at 7.00 am, my average speed is $v$ km/h.
As before, my average speed decreases steadily by 1 km/h for every 2 minutes after 7.00 am that I leave home.
(a) I leave my new home $x$ minutes after 7.00 am.
Show that a model for the time, $T$, minutes, to drive to work is $T = \frac{120d}{2v-x}$.
[2]
(b) I want to leave my new home at 7.30 am and arrive at work at 9.00 am.
Find a model for $v$ in terms of $d$.
............................................. [3]
585
589
600
