Algebra, Year 10: inequalities and sequences

dashed

提示The line is drawn broken up, to show that the points on it are not part of the region.

为什么For y < x + 2, points on the line y = x + 2 itself do not satisfy the inequality, so the boundary is not part of the region and is drawn broken. For ≤ or ≥ the boundary is included and is drawn solid. It matches the open and closed circles used on a number line.

Below the line

提示Pick any x-value and ask how the height of such a point compares with 2x + 1 there.

为什么At each x-value the line is at height 2x + 1, and the inequality wants points whose y-coordinate is less than that, so they sit underneath. For example at x = 1 the line is at height 3, and (1, 0) satisfies 0 < 3.

Test a point that is not on the line, such as (0, 0): 0 + 0 < 6 is true, so it is the side containing the origin

提示Choose an easy pair of coordinates and try them in the inequality.

为什么All the points on one side of a boundary line give the same verdict, so one test settles it. The origin is the easiest point to try unless the line passes through it. If the test point had failed, the region would be the other side.

3

提示Start with the smallest x allowed and list the y-values that work, then move to the next x.

为什么With x = 1, y can be 1 or 2. With x = 2, y can only be 1. With x = 3, y would have to be 0, which is not allowed. So the points are (1, 1), (1, 2) and (2, 1). On a graph they are the grid points inside or on the edge of a small triangle.

x + y < 3 (the same as y < 3 − x)

提示Find the equation of the line first, then use the origin and the style of the line to choose the sign.

为什么The line through (0, 3) and (3, 0) is x + y = 3. At the origin x + y = 0, which is less than 3, so the shaded side is "less than". The line is dashed, so points on it are not included and the sign is <, not ≤.

A quadratic sequence

提示Its nth term contains n², and its name is shared with expressions that contain x².

为什么In 2, 5, 10, 17, 26 the differences are 3, 5, 7, 9, which go up by 2 each time. A linear (arithmetic) sequence has constant first differences; this kind needs a second row of differences before a constant appears.

38

提示Write down the gaps between the terms and look for the pattern in the gaps.

为什么The differences are 3, 5, 7 and 9, which go up by 2 each time, so the next difference is 11 and the next term is 27 + 11. The sequence is not linear, so the last difference cannot simply be reused.

3; 8; 15

提示Put n = 1, then 2, then 3 into the rule, squaring before you add.

为什么n = 1 gives 1 + 2, n = 2 gives 4 + 4 and n = 3 gives 9 + 6. The differences between the terms are 5 and 7, which are not equal, as expected when the rule contains n².

2

提示Find the gaps between the gaps.

为什么5 − 3, 7 − 5 and 9 − 7 are all the same. A constant second difference is the test for a quadratic sequence, and it shows how to continue: the next first difference is 11, so the next term is 40.

Yes: the differences 2, 3, 4, 5 go up by 1 each time, so the second differences are constant

提示Write out the gaps between the terms, then the gaps between those.

为什么The first differences are 2, 3, 4, 5 and the second differences are 1, 1, 1. A constant second difference is what makes a sequence quadratic. The square numbers 1, 4, 9, 16, 25 are another example, with second differences of 2.

A Fibonacci-type sequence

提示It is named after an Italian mathematician who wrote about breeding rabbits.

为什么The original example is 1, 1, 2, 3, 5, 8, 13, … Any two starting numbers can be used with the same rule: 2, 5, 7, 12, 19, … is one as well.

19

提示Look at how 7 and 12 were each made from the terms before them.

为什么2 + 5 = 7 and 5 + 7 = 12, so each term is the sum of the previous two. The next term is 7 + 12.

7

提示The third term was made by adding the first two.

为什么3 + ▢ = 10, so the missing term is 7. Check with the next term: 7 + 10 = 17, as given. Working backwards uses subtraction, because each term is a sum.

a + b, a + 2b and 2a + 3b

提示Each new term adds the two just before it; collect like terms as you go.

为什么Third: a + b. Fourth: b + (a + b) = a + 2b. Fifth: (a + b) + (a + 2b) = 2a + 3b. If you are told the value of two terms, these expressions give equations for a and b.