A person has 2 parents, 4 grandparents, 8 great grandparents and so on. What is the number of his ancestors during the $5^{\text{th}}$ generation preceding his own?
Show Hint
A common mistake is to calculate the total ancestors up to the fifth generation:
\[
2+4+8+16+32=62
\]
However, the question asks for the number of ancestors during the fifth generation only, not the cumulative total. Therefore, the required answer is \(32\), not \(62\).
Step 1: Identify the sequence of ancestors generation-wise
The problem states:
\[
\text{Parents}=2
\]
\[
\text{Grandparents}=4
\]
\[
\text{Great-grandparents}=8
\]
Continuing this pattern:
\[
2,\;4,\;8,\;16,\;32,\ldots
\]
Clearly, every term is twice the preceding term.
Step 2: Determine the first term and common ratio
The first term is:
\[
a=2
\]
The common ratio is:
\[
r=\frac{4}{2}=2
\]
Checking with the next pair:
\[
r=\frac{8}{4}=2
\]
Thus,
\[
a=2,\qquad r=2
\]
Step 3: Find the number of ancestors in the 5th generation
We need the number of ancestors during the \(5^{\text{th}}\) generation preceding the person's own generation.
Using:
\[
a_n=a\,r^{n-1}
\]
Substituting \(a=2\), \(r=2\), and \(n=5\):
\[
a_5=2(B)^{5-1}
\]
\[
a_5=2(B)^4
\]
\[
a_5=2\times16
\]
\[
a_5=32
\]
Therefore, the number of ancestors in the fifth generation is:
\[
{32}
\]
Step 4: Verification through direct doubling
Let us verify by listing generation-wise values:
\[
1^{\text{st}} \text{ generation}=2
\]
\[
2^{\text{nd}} \text{ generation}=4
\]
\[
3^{\text{rd}} \text{ generation}=8
\]
\[
4^{\text{th}} \text{ generation}=16
\]
\[
5^{\text{th}} \text{ generation}=32
\]
The fifth generation indeed contains 32 ancestors.
Step 5: Final Conclusion
The ancestor count follows a geometric progression with common ratio 2. Using either repeated doubling or the GP formula, the number of ancestors in the fifth generation preceding the person's own generation is:
\[
{32}
\]
Hence, the correct answer is:
\[
{\text{Option (C)}}
\]