51 a B1 2 B1 for a valid quadratic sequence M n2 For example, can generate a quadratic sequence: n2 1, 4, 9, 16, 25 ……. b B1 B1 for a clear explanation nth Similarities: you find the term for both types of sequence by looking at the differences between terms. B1 B1 for a clear explanation Differences: in a linear sequence you find the first term by subtracting the difference from the second term. In a quadratic sequence you also have to look at the second differences. This allows you to extend the differences a, b backwards to find the values of c an2 bn c. nth and in the term of + + c B1 B1 for explaining how you woud find the sequence For a linear sequence, just keep on adding 6 each time to give: B1 B1 for cao 2, 8, 14, 20 4) ………..(6n – The term includes 6n because we nth add 6 each time, 6n 4 because (2 – – 6) = –4. B3 B1 for explaining second differences B1 for use of a table or equivalent For a quadratic equation, we build up B1 for correctly finding a quadratic equation with 2 and the series by again having the first 8 as starting terms differences as 6, then choosing a second difference, say 2. This will give a table such as: nth B2 B1 for explaining how you would find the term Start with the two terms in positions = n 1 (2) and = 2 (8) in the sequence. n Put each second difference as 2. Then complete the first differences as shown.
51 a b c For example, n2 can generate a quadratic sequence: 1, 4, 9, 16, 25 ……. n2 Similarities: you find the nth term for both types of sequence by looking at the differences between terms. Differences: in a linear sequence you find the first term by subtracting the difference from the second term. In a quadratic sequence you also have to look at the second differences. This allows you to extend the differences backwards to find the values of a, b and c in the nth term of an2 + bn + c. For a linear sequence, just keep on adding 6 each time to give: 2, 8, 14, 20 ………..(6n – 4) The nth term includes 6n because we add 6 each time, 6n – 4 because (2 – 6) = –4. For a quadratic equation, we build up the series by again having the first differences as 6, then choosing a second difference, say 2. This will give a table such as: Start with the two terms in positions n = 1 (2) and n = 2 (8) in the sequence. Put each second difference as 2. Then complete the first differences as shown. B1 B1 B1 B1 B1 B3 B2 2 B1 for a valid quadratic sequence B1 for a clear explanation B1 for a clear explanation B1 for explaining how you woud find the sequence B1 for cao B1 for explaining second differences B1 for use of a table or equivalent B1 for correctly finding a quadratic equation with 2 and 8 as starting terms B1 for explaining how you would find the nth term M
Higher_Skills_Algebra_MarkScheme_AQA.docx