Question
(a) These are the first four terms of a sequence.
$3$ $10$ $17$ $24$
(i) Write down the next term.
(ii) Write down the term to term rule for continuing the sequence.
(b) These are the first four terms of another sequence.
$16$ $14$ $11$ $7$
Write down the next two terms of this sequence.
▶️Answer/Explanation
(a)(i) $31$
(a)(ii) add $7$
(b) $2$ $-4$
(a)(i)
$
3, \, 10, \, 17, \, 24
$
the difference between consecutive terms
$
10 – 3 = 7, \quad 17 – 10 = 7, \quad 24 – 17 = 7
$
So, to find the next term
$
24 + 7 = 31
$
(a)(ii)
Since each term increases by 7, the term-to-term rule is
$\text{Add 7}$
(b)
$
16, \, 14, \, 11, \, 7
$
the difference between consecutive terms
$
14 – 16 = -2, \quad 11 – 14 = -3, \quad 7 – 11 = -4
$
The differences are -2, -3, -4, so it looks like the sequence is decreasing by increasing negative values
Subtract 5 from the last term
$
7 – 5 = 2
$
Subtract 6 from the next term
$
2 – 6 = -4
$
$
2, \, -4
$
Question
(a) The $n$th term of a sequence is $n^2-3$.
Find the first three terms of this sequence.
(b) These are the first five terms of a different sequence.
$$
\begin{array}{lllll}
2 & 9 & 16 & 23 & 30
\end{array}
$$
Find the $n$th term of this sequence.
▶️Answer/Explanation
(a) −2 1 6
(b) 7n- 5
(a)
$
n^2 – 3
$
When \( n = 1 \)
$
1^2 – 3 = 1 – 3 = -2
$
When \( n = 2 \)
$
2^2 – 3 = 4 – 3 = 1
$
When \( n = 3 \)
$
3^2 – 3 = 9 – 3 = 6
$
First three terms
$
-2, 1, 6
$
Part (b)
$
2, 9, 16, 23, 30
$
difference between consecutive terms
$
9 – 2 = 7, \quad 16 – 9 = 7, \quad 23 – 16 = 7, \quad 30 – 23 = 7
$
common difference is 7,
The first term is 2.
$
\text{nth term} = a + (n-1)d
$
$
\text{nth term} = 2 + (n-1)(7)
$
$
= 2 + 7n – 7
$
$
= 7n – 5
$
Question
The table shows some information about two sequences.
Complete the table.
▶️Answer/Explanation
Ans: 40, –275
To get the 5th term put $n=5$ in both sequence.
sequence A = $60 -4n=60-20=40$
sequence B= $n^2 -300=(5)^2-300=-275$
Question
These are the first five terms of a sequence.
$11$ $18$ $25$ $32$ $39$
Find an expression for the nth term of the sequence.
▶️Answer/Explanation
Ans: \(7n+4\)
$
11, 18, 25, 32, 39
$
the difference between consecutive terms
\( 18 – 11 = 7 \)
\( 25 – 18 = 7 \)
\( 32 – 25 = 7 \)
\( 39 – 32 = 7 \)
The common difference is 7.
The formula for the \( n \)th term of an arithmetic sequence is
$
a_n = a_1 + (n – 1)d
$
Where
\( a_1 \) is the first term (\( 11 \))
\( d \) is the common difference (\( 7 \))
$
a_n = 11 + (n – 1)(7)
$
$
a_n = 11 + 7n – 7
$
$
a_n = 7n + 4
$
Question
(a) Write down the next term in each of these sequences.
(i) 5 9 13 17 . . .
▶️Answer/Explanation
The given sequence is an arithmetic sequence where each term is obtained by adding a constant difference of 4 to the previous term.
So, the next term in the sequence would be:
\(17 + 4 = 21\)
Therefore, the next term in the sequence is 21.
(ii) 3 6 12 24 . . .
▶️Answer/Explanation
The given sequence is a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio of 2.
So, the next term in the sequence would be:
\(24 \times 2 = 48\)
Therefore, the next term in the sequence is 48.
(b) Here are the first four terms in a different sequence.
2 7 12 17
Find an expression for the nth term of this sequence.
▶️Answer/Explanation
The given sequence is an arithmetic sequence where each term is obtained by adding a constant difference of 5 to the previous term.
To find an expression for the nth term of this sequence, we can use the formula for the nth term of an arithmetic sequence:
\(a_n = a_1 + (n – 1) \cdot d\)
Where:
\(a_n\) is the nth term of the sequence.
\(a_1\) is the first term of the sequence.
\(n\) is the term number.
\(d\) is the common difference between consecutive terms.
In this case, the first term \(a_1\) is 2 and the common difference \(d\) is 5.
Substitute the values into the formula:
\(a_n = 2 + (n – 1) \cdot 5\)
\(a_n = 2 + 5n – 5\)
\(a_n = 5n – 3\)
Therefore, the expression for the nth term of the given sequence is \(5n – 3\).
Question
(a) 2, 3, 6, 11, 18, . . .
(i) Write down the next two terms in this sequence.
Answer/Explanation
Ans: 27, 38
(ii) Describe, in words, the rule for continuing this sequence.
Answer/Explanation
Ans: Add the next odd number oe
(b) The nth term of a different sequence is 4n – 3.
Work out the first three terms in this sequence.
Answer/Explanation
Ans: 1, 5, 9
Question
Find the next term in each of these sequences.
(a) 3, 7, 11, 15, …
(b) 10, 7, 4, 1, …
(c) 1, 9, 25, 49, …
Answer/Explanation
Ans:
(a) 19
(b) -2
(c) 81
Question
35, 41, 47, 53, 59,
For this sequence, write down
(a) the next term,
(b) the nth term.
Answer/Explanation
Ans:
(a) 65
(b) 6n + 29 oe