Algebra, Year 10: inequalities and sequences

dashed

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

The whyFor 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

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

The whyAt 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

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

The whyAll 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

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

The whyWith 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)

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

The whyThe 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

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

The whyIn 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

HintWrite down the gaps between the terms and look for the pattern in the gaps.

The whyThe 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

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

The whyn = 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

HintFind the gaps between the gaps.

The why5 − 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

HintWrite out the gaps between the terms, then the gaps between those.

The whyThe 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

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

The whyThe 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

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

The why2 + 5 = 7 and 5 + 7 = 12, so each term is the sum of the previous two. The next term is 7 + 12.

7

HintThe third term was made by adding the first two.

The why3 + ▢ = 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

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

The whyThird: 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.