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INVESTIGATION TWO-STEP SEQUENCES
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 1.
1st term = 1
2nd term = 1st term × 2 + 1 = 1 × 2 + 1 = 3
3rd term = 2nd term × 2 + 1 = 3 × 2 + 1 = 7
4th term = 3rd term × 2 + 1 = 7 × 2 + 1 = 15
(a) Work out the 5th term of this sequence.
................................................... [2]
(b) The 3rd term of this sequence is 7.
You can write 7 as $2^3 - 1$.
Complete the table.
[Table_1: \begin{tabular}{|c|c|c|} \hline 1st term & 1 & 2^1 - 1 \\ \hline 2nd term & 3 & 2^2 - 1 \\ \hline 3rd term & 7 & 2^3 - 1 \\ \hline 4th term & 15 & \\ \hline 5th term & & \\ \hline \end{tabular}] [1]
(c) Calculate the 20th term of this sequence.
Write down all the digits shown on your calculator.
................................................... [2]
(d) (i) Use the last column in the table to write down an expression for the $n^{th}$ term of this sequence.
................................................... [1]
(ii) Show that your expression gives the correct value for the 6th term of this sequence.
................................................... [2]
524
526
558
In this question the two steps are:
• multiply by 3
• then add 4.
The first term is 1.
(a) Calculate the 2nd, 3rd and 4th terms of this sequence.
1, ..........., ..........., ........... [3]
(b) Complete the table.
[Table_1]
| 1st term | 1 | $3^1 - 2$ |
| 2nd term | $3^2 - 2$ | |
| 3rd term | ||
| 4th term | ||
| 5th term | 241 | $3^5 -$ |
(c) Write down an expression for the $n^{th}$ term of this sequence.
........................................... [1]
505
552
505
In this question the two steps are:
- multiply by 4
- then add 9.
The first term is 1.
Show that the expression for the $n^{th}$ term, $4^n - 3$, gives the correct value for the 3rd term of this sequence.
595
550
594
(a) Copy your results from Question 1(d)(i) and Question 2(c) into the table.
Use any patterns you notice to complete the table.
[Table_1]
Question 1(d)(i) Multiply by 2, then add 1 ...............................
Question 2(c) Multiply by 3, then add 4 ...............................
Multiply by 4, then add 9 $4^n - 3$
Multiply by ............, then add 16 $5^n - ..........$
Multiply by 6, then add ........... .......... - 5
Multiply by 7, then add 36 ...............................
Multiply by ............, then add ........... $8^n - 7$ [4]
(b) A sequence has the two steps that you found in the last row of the table.
Show that the expression for the $n^{th}$ term gives the correct value for the 3rd term of this sequence. [3]
(c) The $n^{th}$ term of a two-step sequence is $22^n - 21$.
Find the two steps.
• .................................................................
• .................................................................
(d) In a two-step sequence the steps are:
• multiply by 11
• then add 100.
The first term is 1.
(i) Find the value of the term nearest to 20 000 000.
Write down all the digits shown on your calculator.
................................................... [2]
(ii) Which term in the sequence is your answer to part (i)?
................................................... [1]
549
532
567
In this question the steps in Question 1 are in the reverse order.
The two steps are now:
• add 1
• then multiply by 2.
(a) The first term is 1.
The second term is 4.
Calculate the 3rd, 4th and 5th terms.
1, 4, .......... , .......... , ......... [2]
(b) This two-step sequence has nth term equal to $a \times 2^n - 2$.
(i) The first term is 1.
Use this to find the value of $a$.
............................................. [2]
(ii) Use part (i) to show that the expression for the nth term gives the correct value for the 3rd term of this sequence. [2]
600
563
572
