One example of an arithmetic progression is 16, 20, 24. Use three of the numbers in the grid below to make an arithmetic progression. Describe the rule of the progression that you make. The numbers are eight, 13, 21, 24, 29, and 36.
Let’s firstly consider the arithmetic progression in the example, the numbers 16, 20, 24. We know that these three numbers form an arithmetic progression, as the difference between each term is the same. To get from the first term, 16, to the second term, 20, we add four. To get from the second term, 20, to the third term, 24, we also add four. When we have a common difference, as in this case, we know that we have an arithmetic progression. The rule to get from one term to the next in this arithmetic progression is add four.
To work out the general rule for the sequence or progression, we need to use this information to find the 𝑛th-term formula. The first part of this 𝑛th-term formula of any arithmetic progression is the common difference multiplied by 𝑛, in this case four multiplied by 𝑛 or four 𝑛. Next, we need to consider what sequence the formula four 𝑛 would generate. Four 𝑛 would generate the four times table, or the multiples of four. The first term, when 𝑛 equals one, would be four multiplied by one. This is equal to four. The second term would be eight, as four multiplied by two is equal to eight. This means that the sequence or progression generated by the formula, or rule, four 𝑛 would be four, eight, 12, and so on.
We work out the second part of the formula by calculating how we get from each term of this sequence to each corresponding term in the original sequence. How can we get from four to 16, from eight to 20, and from 12 to 24? This must be an addition or a subtraction. We know that four plus 12 is equal to 16. Eight plus 12 is equal to 20. And 12 plus 12 is equal to 24. This means that the second part of the formula is add 12. The general rule for the arithmetic progression 16, 20, 24 is four 𝑛 plus 12.
We can check this by, once again, substituting in 𝑛 equals one, two, three, or any other integer value. Substituting in 𝑛 equals one gives us four multiplied by one plus 12. This is equal to 16. And the first number in the pattern was 16. Substituting in 𝑛 equals two and 𝑛 equals three gives us the values 20 and 24, respectively. These were the second and third numbers in the progression. As we’ve looked at what is meant by an arithmetic progression and its general rule, now let’s look at this specific question.
We are asked to use three of the numbers in the grid to make an arithmetic progression. In order to do this, they must have the same common difference. As the six numbers are already in order, we can begin by looking at the difference between each of them. The difference between the first number and second number is five, between the second and third is eight. The other differences are three, five, and seven. If any two consecutive numbers were the same, we would have our three numbers.
Unfortunately, as none of them are, we will have to miss a number at some point. Three plus five is equal to eight. This means that the difference between the third number, 21, and the fifth number, 29, is eight. The difference between the second and third number was also eight. We can therefore say that three of the numbers in the grid that make an arithmetic progression are 13, 21, and 29. This is because they have the same common difference of eight. To get from the first term to the second time, we add eight. And to get from the second term to the third term, we also add eight.
As mentioned in the example, this means that the first part of our general rule will be eight 𝑛 or eight multiplied by 𝑛. Eight multiplied by one is equal to eight. Eight multiplied by two is equal to 16. And eight multiplied by three equals 24. How can we get from the numbers in the eight times table to the numbers in our progression, from eight to 13, 16 to 21, and 24 to 29? In all three cases, we need to add five, as eight plus five equals 13, 16 plus five is 21, and 24 plus five is equal to 29.
The second part of our general rule or 𝑛th-term formula is add five. Three numbers from the grid that make an arithmetic progression are 13, 21, and 29. The rule or formula for this progression is eight 𝑛 plus five. Eight multiplied by one plus five is equal to 13. Eight multiplied by two plus five is equal to 21. And eight multiplied by three plus five is equal to 29. Therefore, the first, second, and third terms of the progression are 13, 21, and 29, respectively.