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??? - 2015 JUNE Q10

Jill has 3 daughters and no sons. They are generation 1 of Jill’s descendants.

Each of her daughters has 3 daughters and no sons. Jill’s 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3.

Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.

(i) How many of Jill’s descendants would there be in generation 8? [2]

(ii) How many of Jill’s descendants would there be altogether in the first 15 generations? [3]

(iii) After n generations, Jill would have over a million descendants altogether. Show that n satisfies the inequality
Picture
​Hence find the least possible value of n. [4]

​(iv) How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? [3]
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