Correct Option: B) 9237 Let the given series be denoted by $T_n$, where $T_1 = 3, T_2 = 7, T_3 = 23, T_4 = 119, T_5 = 839$. We aim to find $T_6$. To identify the pattern, we analyze the relationship between consecutive terms: 1. **From $T_1$ to $T_2$:** $T_2 = 7$ $T_1 = 3$ Observe that $7 = 3 \times 2 + 1$. 2. **From $T_2$ to $T_3$:** $T_3 = 23$ $T_2 = 7$ Observe that $23 = 7 \times 3 + 2$. 3. **From $T_3$ to $T_4$:** $T_4 = 119$ $T_3 = 23$ Observe that $119 = 23 \times 5 + 4$. 4. **From $T_4$ to $T_5$:** $T_5 = 839$ $T_4 = 119$ Observe that $839 = 119 \times 7 + 6$. We can generalize this pattern as $T_{n+1} = T_n \times P_n + Q_n$, where $P_n$ represents the multiplier and $Q_n$ represents the adder for the $n$-th step. Let's analyze the sequence of multipliers, $P_n$: $P_1 = 2$ $P_2 = 3$ $P_3 = 5$ $P_4 = 7$ This sequence $2, 3, 5, 7, \dots$ clearly corresponds to the sequence of consecutive prime numbers, starting with the first prime number. Now, let's analyze the sequence of adders, $Q_n$: $Q_1 = 1$ $Q_2 = 2$ $Q_3 = 4$ $Q_4 = 6$ This sequence $1, 2, 4, 6, \dots$ is not a simple arithmetic progression. However, we can identify a pattern: - The first term is $Q_1 = 1$. - For subsequent terms (i.e., for $n \ge 2$), the pattern follows $Q_n = 2(n-1)$. - For $n=2$, $Q_2 = 2(2-1) = 2 \times 1 = 2$. - For $n=3$, $Q_3 = 2(3-1) = 2 \times 2 = 4$. - For $n=4$, $Q_4 = 2(4-1) = 2 \times 3 = 6$. Combining these observations, the recursive pattern for the series is: $T_{n+1} = T_n \times P_n + Q_n$ Where: - $P_n$ is the $n$-th prime number ($P_1=2, P_2=3, P_3=5, P_4=7, P_5=11, \dots$). - $Q_n$ is defined as $Q_1=1$ and $Q_n = 2(n-1)$ for $n \ge 2$. To determine the next term, $T_6$, we need to find $P_5$ and $Q_5$: - The 5th prime number, $P_5$, in the sequence $2, 3, 5, 7, \dots$ is $11$. - Using the formula for $Q_n$ for $n=5$, we get $Q_5 = 2(5-1) = 2 \times 4 = 8$. Now, we substitute these values into the general formula to calculate $T_6$: $T_6 = T_5 \times P_5 + Q_5$ $T_6 = 839 \times 11 + 8$ First, perform the multiplication: $839 \times 11 = 839 \times (10 + 1) = 8390 + 839 = 9229$. Next, perform the addition: $T_6 = 9229 + 8 = 9237$. Thus, the next term in the series is 9237.