Chapter ISG, Problem 7.10.8P

To determine

To write: Both an explicit and recursive formula for the given situation.

Answer to Problem 7.10.8P

Yes, the given situation is possible. Explicit formula is $a_n={(2)}^{n+2}$ and recursive formula is $a_n=2(a_{n-1})$.

Explanation of Solution

Given information: Situation is regarding great-great-great-great-great-great grandparents.

As the occurrence of word ‘great’ before word ‘grandparents’ increases, the total number of them also increases, consider a single grandparent to be a node which is connected to two more nodes as one goes up in the family tree. Note that occurrence of word great one time means $8$ members, two occurrences of word ‘great’ means $8\times 2=16$ members and so on.

Explicit formula:

  $a_n={(2)}^{n+2}$ , where $n$ is the number of times the word ‘great’ occurs.

Putting $n=6$ ,

  $a_6={(2)}^8=256$.

So, the given situation is possible.

Recursive Formula:

  $a_n=2(a_{n-1})$ , where $n$ is the number of times the word ‘great’ occurs.

Thus, explicit formula is $a_n={(2)}^{n+2}$ and recursive formula is $a_n=2(a_{n-1})$.