**??? - 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

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 inequalityHence find the least possible value of

(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]

*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]