Question

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?

• A. 40
• B. 124
• C. 32
• D. 62

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 \(62\).

Answer 1

The Correct Option is C

Concept: Each generation backward doubles the number of ancestors because every individual has exactly two biological parents. Thus, the number of ancestors forms a geometric progression (G.P.): \[ 2,\;4,\;8,\;16,\;32,\ldots \] A geometric progression is a sequence in which each term is obtained by multiplying the previous term by a fixed number called the common ratio. The general term of a geometric progression is: \[ a_n=a\,r^{\,n-1} \] where:
• \(a\) = first term
• \(r\) = common ratio
• \(n\) = term number

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: \[ \boxed{32} \]

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: \[ \boxed{32} \] Hence, the correct answer is: \[ \boxed{\text{Option (C)}} \]