Question

Find the missing number: 2, 6, 20, 42, 110, ____

• A. 126
• B. 156
• C. 176
• D. 196

Hint:

Write each number as a product of two consecutive integers, then notice the smaller integer in each pair is always one less than a prime number.

Answer 1

The Correct Option is B

Step 1: Reframe each term directly using prime numbers, without splitting by trial.
Check if each term equals $p \times (p-1)$ for some prime $p$: for $p=2$, $2 \times 1 = 2$. For $p=3$, $3 \times 2 = 6$. For $p=5$, $5 \times 4 = 20$. For $p=7$, $7 \times 6 = 42$. For $p=11$, $11 \times 10 = 110$. Every given term fits this rule using the primes $2, 3, 5, 7, 11$ in order.

Step 2: The sequence is built from consecutive primes.
So the $n$th term of this series is $p_n(p_n-1)$, where $p_n$ is the $n$th prime number. We are given the first five terms, so the sixth term needs the sixth prime.

Step 3: Identify the sixth prime and compute.
The sixth prime number is 13. Term = $13 \times (13-1) = 13 \times 12 = 156$.

Final Answer:
The missing number is $156$. \[ \boxed{156} \]