For the following sequences, write down the next two terms:
4, 8, 12, 16, \ldots
2, 3.5 , 5 , 6.5, \ldots
6, 2, - 2, - 6, \ldots
\dfrac{3}{4}, \dfrac{2}{4}, \dfrac{1}{4}, \dfrac{0}{4}, \ldots
State whether the following sequences represent an arithmetic progression:
3, 6, 12, 24 ...
5, 7, 5, 7 ...
Find the common difference, d, for the following arithmetic sequences:
- 6, - \dfrac{39}{7}, - \dfrac{36}{7}, - \dfrac{33}{7}, \ldots
330, 280, 230, 180, \ldots
3, 0, - 3 , - 6 , \ldots
A sequence with first term 3, and the fifth term 19
Write the first four terms in the following arithmetic sequences:
The first term is - 10 and the common difference is 4.
The first term is - 8 and the common difference is - 2.
The first term is a and the common difference is d.
Find the missing terms in the following arithmetic sequences:
8, x, 16, 20, y
a, 0, b, 10, c
Consider the sequence defined by a_1 = 6 and a_n = a_{n - 1} + 5 for n \geq 2. Find:
The 21st term
The 22nd term
The 23rd term
The 24th term
For the following sequences:
Write a recursive rule for T_n in terms of T_{n - 1} which defines the sequence, and an initial condition for T_1.
Find the 100th term in the sequence.
Each term is obtained by increasing the previous term by 15. The first term is 20.
The first term of an arithmetic sequence is 3. The fifth term is 19.
Each term is obtained by decreasing the previous term by 5. The first term is 2.
7, 17, 27, 37, \ldots
For the following sequences, write a recursive rule for T_{n + 1} in terms of T_n and the initial condition for T_1:
Each term is obtained by increasing the previous term by 35. The first term is 60.
Each term is obtained by the previous term minus 4. The first term is 2.
Consider the arithmetic sequence with terms T_3 = 14 and T_{12} = 59.
Find d, the common difference.
Find the first term, T_1.
Write a recursive rule for T_n in terms of T_{n - 1} and the initial condition for T_1.
For each of the following general rules, find:
The first four terms of the sequence.
The common difference in the sequence.
A recursive rule for the sequence, and an initial condition for T_1.
Consider the arithmetic sequence 9, 15, 21, \ldots
Find d.
State the general rule for T_n.
Find T_9.
Find the 90th term in the sequence.
For the following sequences:
Write the general rule for the nth term of the sequence, a_n, in terms of n.
Find the 90th term in the sequence.
a_1 = - 17, a_n = a_{n - 1} - 8 for n \geq 2
8.5, 3, - 2.5 , - 8 , \ldots
For the following arithmetic sequences:
Find d.
State the equation for finding T_n.
Find the indicated term in brackets.
17, 16.2, 15.4, \ldots (T_{13})
10, 3, - 4, \ldots (T_9)
5, \dfrac{23}{4}, \dfrac{13}{2}, \ldots (T_{14})
For the following formulas of arithmetic progressions:
Find a.
Find d.
Find the indicated term in brackets.
T_n = 15 + 5 \left(n - 1\right), (T_9)
T_n = - 8 n + 28, (T_5)
Find the five consecutive terms of an arithmetic sequence that sit between - 12 and 24 as shown:
- 12, a, b, c, d, e, 24
In an arithmetic progression, T_{11} = 27 and T_{13} = 31.
Form an equation for a in terms of d, using the fact that T_{11} = 27.
Form another equation for a in terms of d using the fact that T_{13} = 31.
Hence, find the value of d.
Hence find the value of a.
Find T_{10}.
In an arithmetic progression, T_5 = \dfrac{51}{5} and T_{15} = \dfrac{91}{5}.
Form an equation for a in terms of d, using the fact that T_5 = \dfrac{51}{5}.
Form another equation for a in terms of d using the fact that T_{15} = \dfrac{91}{5}.
Hence, find the value of d.
Hence, find the value of a.
Find T_{27}.
In an arithmetic progression where a is the first term, and d is the common difference, we have T_5 = 26 and T_{10} = 51.
Find the value of d.
Find the value of a.
State the general rule T_n.
Hence find T_{25}.
Calculate the sum of the first 25 terms.
In an arithmetic progression, T_5 = 15 and T_{20} = 45.
Find the value of d.
Find the value of a.
Find T_{10}.
Calculate the sum of the first 11 terms.
In an arithmetic progression, T_4 = 5.2 and T_{14} = 9.2.
Find the value of d.
Find the value of a.
Find T_{26}.
Calculate the sum of the first 25 terms.
An arithmetic progression with first term a and common difference d has 13 terms.
The 6th term is - 2. Form an equation for a in terms of d.
The 7th term is - 4. Form another equation for a in terms of d.
Hence find the value of d.
Hence find the value of a.
Hence find the last term in the progression.
In an arithmetic progression, the first term is 32.
Using T_n = a + \left(n - 1\right) d, write an expression for the 5th term.
Using T_n = a + \left(n - 1\right) d, write an expression for the 9th term.
Given that the 9th term is 4 times the 5th term, find the common difference d.
The first three terms of an arithmetic progression are given below.
46, 41, 36, \ldots
Using the fact that the nth term is given by T_n = a + \left(n - 1\right) d, find the range of values of n for which the terms in the progression are positive.
What is the last positive term in the progression?
Find the value of x such that x + 4, 4 x + 4, and 10 x - 20 form successive terms in an arithmetic progression.
The 14th term of an arithmetic progression is equal to the sum of the 5th term and the 10th term. The common difference is - 2.
Find an expression for T_{14} in terms of a.
Find the value of a.
Hence, find the 9th term.
Determine the number of terms, n, for each of the following sequences:
0.9, 1.5, 2.1, \text{. . .}, 22.5
2, 7, 12, \text{. . .}, 132
2, - 3 , - 8 , \text{. . .}, -578
5, \dfrac{17}{4}, \dfrac{7}{2}, \text{. . .}, -37
Given that T_1 = 28 and d = - 18 for an arithmetic sequence, find n if T_n = - 530.
For each of the following tables containing the terms of an arithmetic sequence, find:
d
T_n
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 8 | 13 | 18 | 23 |
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 6 | 1 | -4 | -9 |
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 6 | \dfrac{33}{4} | \dfrac{21}{2} | \dfrac{51}{4} |
| n | 1 | 3 | 5 | 7 |
|---|---|---|---|---|
| T_n | -1 | -21 | -41 | -61 |
| n | 1 | 4 | 10 |
|---|---|---|---|
| T_n | 7 | -17 | -65 |
For the following first-order recurrence relationships:
Determine the next four terms of the sequence, from T_2 to T_5.
Plot the first five terms on a number plane.
State whether the sequence generated is arithmetic.
T_n = T_{n - 1} - 2, T_1 = 5
Consider the sequence plot drawn:
State the first five terms of the sequence.
Is the sequence arithmetic?
Write a recursive rule for t_n in terms of t_{n - 1} and an initial condition for t_1.
Consider the sequence 4, 6, 8, 10, 12, \ldots
Plot the sequence on a number plane.
Is this sequence arithmetic?
Write a recursive rule for t_n in terms of t_{n - 1} and an initial condition for t_1.
The values in the table show the nth term in an arithmetic sequence for consecutive values of n. Complete the missing values in the table:
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n | -6 | -26 |
The nth term of an arithmetic progression is given by T_n = 12 + 4 \left(n - 1\right).
Complete the given table of values:
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
Plot the points in the table on a number plane.
If the points on the graph were joined, would they form a straight line or a curve?
By how much are the consecutive terms in the sequence increasing by?
The nth term of an arithmetic progression is given by T_n = 15 - 5 \left(n - 1\right).
Complete the given table of values:
| n | 1 | 2 | 3 | 4 | 20 |
|---|---|---|---|---|---|
| T_n |
Plot the points in the table on a number plane.
If the points on the graph were joined, would they form a straight line or a curve?
What is the difference between consecutive terms?
Consider the following graph of points:
Complete the table of values.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n |
Find the value of d.
Determine the general term of the sequence, T_n.
Find the 18th term of the sequence.
For the following graph of points:
Complete the table of values.
| n | 1 | 3 | 5 |
|---|---|---|---|
| T_n |
Find the value of d.
Determine the general term of the sequence, T_n.
Find the 18th term of the sequence.
The plotted points represent terms in an arithmetic sequence:
Find the common difference, d.
Determine the general term of the sequence, T_n.
If a line were drawn through the plotted points, determine the gradient of the line.
The plotted points below represent terms in an arithmetic sequence:
Find the common difference, d.
Determine the general rule of the sequence, T_n.
The points are reflected across the horizontal axis to form three new points.
If these new points represent consecutive terms of an arithmetic sequence, write the general rule for A_n, the nth term in this new sequence.