Correct Answer: (A) Let's analyze the given number series: `5, 12, 26, 54, 110, \dots` We need to identify the underlying pattern or rule governing the sequence of terms. Let $T_n$ denote the $n$-th term of the series. Step 1: Examine the relationship between consecutive terms. For the first two terms: $T_2 = 12$ $T_1 = 5$ We can observe that $12 = 2 \times 5 + 2$. For the second and third terms: $T_3 = 26$ $T_2 = 12$ We can observe that $26 = 2 \times 12 + 2$. For the third and fourth terms: $T_4 = 54$ $T_3 = 26$ We can observe that $54 = 2 \times 26 + 2$. For the fourth and fifth terms: $T_5 = 110$ $T_4 = 54$ We can observe that $110 = 2 \times 54 + 2$. Step 2: Establish the general rule. From the observations in Step 1, it is clear that the pattern for the series is: $T_n = 2 \times T_{n-1} + 2$ where $T_n$ is the $n$-th term and $T_{n-1}$ is the $(n-1)$-th term. Step 3: Calculate the next term. To find the next term, which is the sixth term ($T_6$), we apply the established rule using the fifth term ($T_5 = 110$): $T_6 = 2 \times T_5 + 2$ $T_6 = 2 \times 110 + 2$ $T_6 = 220 + 2$ $T_6 = 222$ Alternative Method (Differences): One could also observe the differences between consecutive terms: * $12 - 5 = 7$ * $26 - 12 = 14$ * $54 - 26 = 28$ * $110 - 54 = 56$ These differences form a Geometric Progression: `7, 14, 28, 56, \dots`, where each term is twice the previous term. The next difference in this sequence would be $56 \times 2 = 112$. To find the next term in the original series, we add this difference to the last known term: Next term = $110 + 112 = 222$. Both methods lead to the same result. Step 4: Analysis of other options. * **(B) 220:** This option would be correct if the pattern was simply $T_n = 2 \times T_{n-1}$ (i.e., neglecting the addition of 2). For example, $2 \times 110 = 220$. * **(C) 218:** This option would arise from an incorrect application of the rule, for instance, if one mistakenly calculated $2 \times 110 - 2 = 218$. * **(D) 224:** This option would be correct if the pattern was $T_n = 2 \times T_{n-1} + 4$ (i.e., adding 4 instead of 2). Therefore, the correct term is 222. The final answer is $\boxed{\text{222}}$