Algebra, Year 11: quadratic inequalities and sequences

−2 ≤ x ≤ 4

提示Find where the expression equals zero, then decide which side of those values you need.

为什么x² − 2x − 8 = (x + 2)(x − 4), which is zero at x = −2 and x = 4. The graph is U-shaped, so it is on or below the x-axis between the roots. Test x = 0: −8 ≤ 0, true.

x < −3 or x > 3

提示Sketch y = x² and the line y = 9.

为什么x² = 9 at x = −3 and x = 3. The curve y = x² is above 9 outside these values, so x < −3 or x > 3. Test x = −4: 16 > 9, true. Test x = 0: 0 > 9, false.

{x : x < −3} ∪ {x : x > 3}

提示Each inequality gets its own pair of curly brackets, and the two are joined by the symbol that stands for 'or'.

为什么Curly brackets mean 'the set of', and the colon is read 'such that'. ∪ is the union: everything in either set, which matches the word 'or'. {x : x < −3 or x > 3} says the same thing and is also accepted; a single interval such as 2 < x < 3 is written {x : 2 < x < 3}.

below

提示Less than zero means y is negative.

为什么'< 0' asks where y is negative, which is where the graph dips under the axis: between the roots, 2 < x < 3. For '> 0' you would want the parts above the axis, x < 2 or x > 3. A quick sketch settles which.

7

提示Remember zero and the negative whole numbers.

为什么x² < 10 when x is between −√10 and √10, about −3.16 to 3.16. The integers in that range are −3, −2, −1, 0, 1, 2 and 3, which is 7 of them.

12

提示Going from the 1st term to the 5th takes four multiplications.

为什么5th term = 3 × (√2)⁴. Since (√2)² = 2, (√2)⁴ = 4, so the term is 3 × 4 = 12. The sequence is 3, 3√2, 6, 6√2, 12.

6√3

提示Divide one term by the term before it to find the multiplier.

为什么2√3 ÷ 2 = √3, and 2√3 × √3 = 2 × 3 = 6, so the common ratio is √3. The next term is 6 × √3 = 6√3, and the one after that is 18.

8

提示Apply the rule twice, one step at a time.

为什么u₂ = 2² − 1 = 3 and u₃ = 3² − 1 = 8. Rules like this, where each term is worked out from the one before, are given in the question; the skill is to apply them carefully, one term at a time.

n² on its own already has a second difference of 2, so the coefficient of n² is half the second difference

提示Look at how the gaps between 1, 4, 9 and 16 grow.

为什么The square numbers 1, 4, 9, 16 have differences 3, 5, 7, which go up by 2 each time. Multiplying by a multiplies that 2 by a, so an² has second difference 2a. Here 2a = 6, so a = 3.

n² + 1

提示Compare each term with the square numbers.

为什么The differences are 3, 5, 7, 9, so the second difference is 2 and the nth term starts with n². Subtracting the square numbers 1, 4, 9, 16, 25 leaves 1 every time, so the nth term is n² + 1.

4; 2; n + 5

提示Find the differences of the differences, then halve.

为什么First differences: 7, 11, 15, 19; second differences: 4. Half of 4 is 2, so the rule starts 2n², which gives 2, 8, 18, 32, 50. Taking these away leaves 6, 7, 8, 9, 10, the sequence n + 5, so the nth term is 2n² + n + 5.

3

提示Put n = 2 into the rule and set the result equal to 10.

为什么When n = 2, n² + bn = 4 + 2b. Setting 4 + 2b = 10 gives b = 3. The sequence is then 4, 10, 18, 28, …

2n² − n

提示Halve the second difference, then see what is left once that part is taken away from each term.

为什么First differences: 5, 9, 13, 17; second differences: 4, so the rule starts 2n², giving 2, 8, 18, 32, 50. Subtracting these from the terms leaves −1, −2, −3, −4, −5, which is −n. So the nth term is 2n² − n.