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14 The sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 gives the powers of 2, less than 1000.

a Write the sequence of powers of 3 less than 1000 (starting 1, 3, 9,…) b Write the sequence of powers of 5 less than 1000 (starting 1, 5, 25,…) c i For the original sequence (of powers of 2), list the difference between each term and the next term (for example: 2 1 = 1, 4 2 = 2,…) ii What do you notice about this sequence of differences? d i List the difference between consecutive terms in the powers of 3 sequence. ii What do you notice about this sequence of differences?

15 Find a value for a and for b such that a ≠ b and a b = ba

ENRICHMENT: Investigating factorials

16 In mathematics, the exclamation mark (!) is the symbol for factorials. 4! = 4 × 3 × 2 × 1 = 24 a Evaluate 1 !, 2 !, 3 !, 4 !, 5 ! and 6 !

Factorials can be written in prime factor form, which involves powers. For example: b Write these numbers in prime factor form. c Write down the last digit of 12 ! d Write down the last digit of 99 !

16 e Find a method of working out how many consecutive zeros would occur on the right-hand end of each of the following factorials if they were evaluated. (Hint: Consider prime factor form.) i 5 ! ii 6 ! f 10 != 3 !× 5 !× 7 ! is an example of one factorial equal to the product of three factorials. Express 24 ! as the product of two or more factorials.

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