Determine whether each of the following sequences is an arithmetic progression:
1, \sqrt{5}, 5, 5 \sqrt{5}, \ldots
2, 2^{2}, 2^{4}, 2^{6}, \ldots
2, 0, - 2, - 4, \ldots
3, - 3, 3, - 3, \ldots
How is the common difference of an arithmetic sequence obtained?
Find the common difference of the following arithmetic sequences:
2, 0, - 2, - 4, \ldots
330, 280, 230, 180, \ldots
- 6, - \dfrac{39}{7}, - \dfrac{36}{7}, - \dfrac{33}{7}, \ldots
Write down the next two terms of the following arithmetic sequences:
6, 2, - 2, - 6, \ldots
Find the missing terms in the following arithmetic progressions:
8,⬚,16, 20, ⬚
⬚, 0, ⬚, 10, ⬚
For each of the following definitions of the nth term of a sequence:
List the first four terms of the sequence.
Find the common difference.
For each of the following, write the first four terms in the arithmetic progression.
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 t_1 and the common difference is d.
Consider the arithmetic sequence 1.4, 2.3, 3.2, \ldots, 10.4
Determine d, the common difference.
Find n, the number of terms in the sequence.
Insert five terms in the arithmetic sequence which has - 12 as its first term and 24 as its last term:
- 12,⬚, ⬚, ⬚, ⬚, ⬚,24
Find the value of x such that x + 4, 6 x + 5, and 9 x - 8 form successive terms in an arithmetic progression.
An arithmetic progression has T_5 = 21 and T_{19} = 77.
Find the value of d.
Find the value of T_1.
Find T_{11}, the 11th term in the sequence.
An arithmetic progression has T_6 = -11 and T_8 = -17.
Find the value of d.
Find the value of T_1.
Hence find the 14th term in the progression.
In an arithmetic progression, the first term is 32. 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: 34, 27, 20, . . .
Find the range of values of n for which the terms in the progression are positive.
Hence determine the number of positive terms in the progression.
Find the last positive term in the progression.
In an arithmetic progression the 6th term is x and the 10th term is y.
Form an expression for d in terms of x and y.
Form an expression for t_1 in terms of x and y.
Hence, find an expression for the 16th term in terms of x and y.
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 the value of t_1.
Hence, find the 9th term.
Consider the sequence 2, 7, 12, \ldots. If the nth term is 132, find the value of n.
Consider the sequence 2, - 3 , - 8 , \ldots. If the nth term is - 578, find the value of n.
Each term of a sequence is obtained by increasing the previous term by 25. The first term is 30. Write a recursive rule for T_{n+1} in terms of T_n, and an initial condition for T_1 for this sequence.
The first term of an arithmetic sequence is 2. The fifth term is 26.
Find d, the common difference of the sequence.
Write a recursive rule for T_{n+1} in terms of T_n which defines this sequence and an initial condition for T_1.
For each of the following sequences:
Determine d, the common difference.
State the expression for finding T_n, the nth term in the sequence.
Determine T_{10}, the 10th term in the sequence.
For the each of the following formulas for the nth term of an arithmetic progression:
Find T_1, the first term of the sequence.
Find d, the common difference.
Find T_5, the 5th term in the sequence.
Consider an arithmetic progression where T_7 = 44 and T_{14} = 86.
Determine d, the common difference.
Determine T_1, the first term in the sequence.
State the equation for T_n, the nth term in the sequence.
Hence find T_{25}, the 25th term in the sequence.
The given table of values represents terms in an arithmetic sequence:
Find d, the common difference.
Write an expression for the nth term of the sequence, T_n.
Find the 15th term of the sequence.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 9 | 15 | 21 | 27 |
The given table of values represents terms in an arithmetic sequence:
Find d, the common difference.
Write an expression for the nth term of the sequence, T_n.
Find the 10th term of the sequence.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 5 | -4 | -13 | -22 |
The given table of values represents terms in an arithmetic sequence:
Find d, the common difference.
Write an expression for the nth term of the sequence, T_n.
Find the 11th term of the sequence.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n | 4 | \dfrac{17}{2} | 13 | \dfrac{35}{2} |
Consider the sequence 4, 6, 8, 10, 12, \text ...
Plot the points on a graph.
Is this sequence arithmetic or geometric? Explain your answer.
Write a recursive rule for t_{n+1} in terms of t_n and an initial condition for t_1.
The nth term of an arithmetic progression is given by the equation T_n = 15 - 5 \left(n - 1\right).
Complete the table of values:
What is the difference between consecutive terms?
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| T_n |
Plot the points from the table on a number plane.
What type of function will be formed if the points on the graph were joined?
The nth term of an arithmetic progression is given by the equation T_n = 12 + 4 \left(n - 1\right).
Complete the table of values:
By how much are consecutive terms in the sequence increasing?
Plot the points in the table on the graph.
What will be formed if the points on the graph were joined?
| n | 1 | 2 | 3 | 4 | 10 |
|---|---|---|---|---|---|
| T_n |
Consider the first-order recurrence relationship defined by T_{n+1} = T_n + 2, T_1 = 5.
Determine the next four terms of the sequence, from T_2 to T_5.
Plot the first five terms on a graph.
Is the sequence generated from this definition arithmetic or geometric? Explain your answer.
Consider the sequence plot shown:
State the first five terms of the sequence.
Is the sequence depicted by this graph arithmetic or geometric?
Write a recursive rule for t_{n+1} in terms of t_n and an initial condition for t_1.
The plotted points represent terms in an arithmetic sequence:
Complete the table of values for the given points:
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| T_n |
Find d, the common difference.
Write an expression for the nth term of the sequence, T_n.
Find the 14th term of the sequence.
The plotted points represent terms in an arithmetic sequence:
Find d, the common difference.
Write an expression for the nth term of the sequence, T_n.
If a line were drawn through the plotted points, what would be the gradient of the line?
The plotted points represent terms in an arithmetic sequence:
Find d, the common difference.
Write an expression for the nth term of the sequence, T_n.
The points are reflected about the horizontal axis to form three new points. If these new points represent consecutive terms of an arithmetic sequence, write the equation for T_k, the kth term in this new sequence.
A termite treatment will cost \$250 for the first half hour, \$245 for the second half hour, \$240 for the third half hour and so on. Find the cost of a treatment that takes 6 hours.
A ball starts rolling down a slope. It rolls 25 cm during the first second, 53 cm during the second second, 81 cm during the third second and so on. At this rate, what is the total distance it will have rolled after 10 seconds?
A diving vessel descends below the surface of the water at a constant rate so that the depth of the vessel after 1 minute, 2 minutes and 3 minutes is 50 metres, 100 metres and 150 metres respectively.
By how much is the depth increasing each minute?
What will the depth of the vessel be after 4 minutes?
Continuing at this rate, what will be the depth of the vessel after 10 minutes?
In a study, scientists found that the more someone sleeps, the quicker their reaction time. The attached table shows the findings:
| \text{Number of hours of sleep }(x) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Reaction time in seconds } (y) | 6 | 5.8 | 5.6 | 5.4 | 5.2 | 5 |
How much does the reaction time change for each extra hour of sleep?
Write an algebraic equation relating the number of hours of sleep, x, and the reaction time, y.
What would be the reaction time for someone who has slept 4.5 hours?
Predict the number of hours someone sleeps if they have a reaction time of 5.5 seconds.
When a new school first opened, T_1 students started at the school. Each year, the number of students increases by the same amount, d.
At the beginning of its 7th year, it had 361 students. At the end of the 11th year, the school had 536 students.
Find d, the number of students who joined the school each year.
Find T_1, the number of students that started at the school when it first opened.
At the end of the nth year, the school has reached its capacity at 921 students. Find the value of n.
The value of Beirut Bank shares is decreasing by \$2.05 each day. At the beginning of today's trading, the shares are worth \$42.54.
Today is March 8. How much are they worth at the start of March 17?
Write a recursive rule for V_{n+1} in terms of V_n which defines the the value of the shares at the end of day n, and an initial condition for V_1.
The value of an investment that pays simple interest each year is graphed, where V_n is the value of the investment, in dollars, after n years:
Find the value of V_1.
Find the value of d, the amount of interest earned each year.
Write an explicit rule for V_n.
What will the investment be worth after 18 years?
A car bought at the beginning of 2009 is worth \$1500 at the beginning of 2015. The value of the car has depreciated by a constant amount of \$50 each year since it was purchased.
What was the car purchased for in 2009?
Plot the value of the car, V_n, on a graph from 2009, where n = 0, to 2015, where n = 6.
Write an explicit rule for the value of the car after n years.
Find the year in which the car will be worth half the price it was bought for.
A piece of jewellery appreciates in value by a constant amount each year. Its value is modelled by the recurrence relation V_{n+1} = V_n + 320, V_0 = 2\,000, where V_n is the value of the jewellery, in dollars, after n years.
State the initial value of the piece of jewellery.
By how much does it appreciate each year?
Write an explicit rule for V_n that gives the value of the piece of jewellery after n years.
What will the investment be worth after 11 years?
A motorbike depreciates in value by a constant amount each year. Its value is modelled by the recurrence relation V_{n+1} = V_n - 1200, V_0 = 15\,000, where V_n is the value of the motorbike, in dollars, after n years.
State the purchase price of the motorbike.
At what rate is it decreasing in value each year?
Write an explicit rule for V_n that gives the value of the motorbike after n years.
What will the motorbike be worth after 9 years?
The balance of a savings account earning simple interest each year is given by the explicit rule V_n = 2\,200 + 300 \left(n - 1\right), where V_n is the balance after n years.
How much interest is the account earning each year?
How much is in the account after 1 year?
What was the original investment amount?
Write a recursive rule for V_{n+1} in terms of V_n .
A mobile phone depreciates in value by a constant amount per month. Its value is given by the explicit rule V_n = 1\,200 - 20 n, where V_n is the balance after n months.
By how much does the value of the phone depreciate each month?
What was the purchase price of the phone?
Write a recursive rule for V_{n+1} in terms of V_n, and an initial condition V_0.
A rare figurine was purchased for \$60 and ten years later it is worth \$460.
If the figurine appreciated in value at a constant rate, how much did it appreciate by each year?
Write a recursive rule for V_{n+1} in terms of V_n, and an initial condition V_0.
Write an explicit rule, V_n, for the value of the figurine after n years.
What will the value of the figurine be in another 10 years time?
A piece of machinery depreciated at a constant rate per hour of use. After 140 hours of use, it was worth \$28\,300. After 190 hours of use, it was worth \$28\,050.
What was the amount of depreciation each hour?
Find V_0, the initial value of the machinery.
Write a recursive rule for V_{n+1} in terms of V_n, and an initial condition V_0.
Write an explicit rule, V_n, for the value of the machinery after n hours of use.
Find the number of hours of use after which the machinery will be a quarter of its original value.
The recurrence relation which models the value of the investment after {n + 1} months is given by V_{n + 1} = V_n + 11.5, V_0 = 1150.
Is the investment type simple interest or compound interest?
What is the annual interest rate?
Determine the balance of the account after 8 years.
Manpreet lives in India and invests 58\,000 INR into an investment account that pays 6.6\% simple interest per annum.
By what amount will the account increase each year?
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth year and t_0 is the initial investment.
Write the explicit rule that can be used to find the balance at the end of n years.
Determine the balance after 5 years.
Determine how many whole years it takes for the balance to exceed 102\,022 INR.
Sunil lives in India and invests 54\,000 INR into an investment account that pays 5.6\% simple interest per annum.
By what amount will the account increase each year?
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth year and t_0 is the initial investment.
Write the explicit rule that can be used to find the balance at the end of n years.
Determine the balance after 4 years.
Determine how many whole years it takes for the balance to exceed 94\,824 INR.
A retiree invests \$570\,500 into an investment account that pays 6\% simple interest per annum.
By what amount will the account increase each year?
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth year and t_0 is the initial investment.
Write the explicit rule that can be used to find the balance at the end of n years.
Determine the balance after 13 years.
Determine how many whole years it takes for the balance to exceed 1.5 million dollars.
An investor deposits \$74\,000 into an investment account that pays 3.7\% simple interest per annum.
By what amount will the account increase each year?
Write the recurrence relation for this situation, where t_n is the balance at the end of the nth year and t_0 is the initial investment.
Write the explicit rule that can be used to find the balance at the end of n years.
Determine the balance after 9 years.
Determine how many whole years it takes for the balance to exceed \$116\,439.
An investor deposits \$51\,000 into a high risk fund that pays 0.3\% simple interest per month.
By what amount will the investment fund increase each month?
Write the recurrence relation for this situation where n is the number of months.
Write the explicit rule that can be used to find the balance at the end of n months.
Determine the balance after 9 years.
Determine how many whole months it takes for the balance to exceed \$54\,289.5.