The first four terms of a sequence are shown in the grid: negative two, seven, negative three, two. Each term after the third term is found by adding the previous three terms together. Part a) Find the next two terms in the sequence.
There is also a part b) that we will look at later. We’re told that each term is found by adding the three previous terms together. We already have the first four terms of the sequence. We can work out the fifth term of the sequence by adding the second, third, and fourth terms. We need to add seven, negative three, and two. Seven add negative three is equal to four. Adding two to this gives us six. Therefore, the fifth term is six.
As we were asked to work out the next two terms, we also need to work out the sixth. This can be calculated by adding the third, fourth, and fifth terms. We need to add negative three, two, and six. Negative three plus two is equal to negative one. And adding six to this gives us an answer of five. The sixth term in the sequence is five. The next two terms in the sequence from the grid are six and five.
The second part of the question says the following. b) The sequence continues. How many negative terms are there in the sequence? Circle your answer. Is it one, two, three, four, or more than four? Give a reason for your answer.
The fourth, fifth, and sixth numbers in the sequence are two, six, and five. These are all positive numbers. The sum of three positive numbers will always be positive. This means that the seventh number in the sequence will be positive. This pattern will then continue and every other number in the sequence will be positive. Therefore, the only negative terms in the sequence will be the first term, negative two, and the third term, negative three.
We can therefore conclude that the correct answer is two. There are two negative terms in the sequence. The reason for this is that the three terms preceding the seventh term are all positive. As the sum of three positive numbers is positive, all subsequent terms will also be positive.