The correct answer is **C) $248$**. Here's a step-by-step breakdown of the pattern: 1. **Analyze the first layer of differences between consecutive terms:** * $12 - 5 = 7$ * $27 - 12 = 15$ * $58 - 27 = 31$ * $121 - 58 = 63$ The new series of differences is: $7, 15, 31, 63$ 2. **Analyze the second layer of differences (differences between the terms of the first difference series):** * $15 - 7 = 8$ * $31 - 15 = 16$ * $63 - 31 = 32$ The second layer of differences is: $8, 16, 32$ 3. **Identify the pattern in the second layer of differences:** This series ($8, 16, 32$) is a Geometric Progression (GP) where each term is twice the previous term (common ratio = 2). * $8 \times 2 = 16$ * $16 \times 2 = 32$ 4. **Determine the next term in the second layer of differences:** Following the pattern, the next term will be $32 \times 2 = 64$. 5. **Determine the next term in the first layer of differences:** Add the newly found second difference to the last term of the first difference series: $63 + 64 = 127$ 6. **Determine the next term in the original series:** Add the newly found first difference to the last term of the original series: $121 + 127 = 248$ Thus, the next term in the series is $248$. **Step-by-step derivation:** Original Series: $5, 12, 27, 58, 121, ?$ **Step 1: Calculate the first differences (Difference 1)** $12 - 5 = 7$ $27 - 12 = 15$ $58 - 27 = 31$ $121 - 58 = 63$ Series of Difference 1: $7, 15, 31, 63, ?_1$ **Step 2: Calculate the second differences (Difference 2)** $15 - 7 = 8$ $31 - 15 = 16$ $63 - 31 = 32$ Series of Difference 2: $8, 16, 32, ?_2$ **Step 3: Identify the pattern in Difference 2** The series $8, 16, 32$ is a geometric progression where each term is multiplied by $2$ to get the next term. $8 \times 2 = 16$ $16 \times 2 = 32$ **Step 4: Extend Difference 2** The next term in Difference 2 ($?_2$) will be $32 \times 2 = 64$. Series of Difference 2: $8, 16, 32, 64$ **Step 5: Extend Difference 1** The next term in Difference 1 ($?_1$) is found by adding the last term of Difference 1 to the next term of Difference 2: $?_1 = 63 + 64 = 127$ Series of Difference 1: $7, 15, 31, 63, 127$ **Step 6: Determine the next term in the original series** The next term in the original series is found by adding the last term of the original series to the next term of Difference 1: Next term $= 121 + ?_1 = 121 + 127 = 248$ Therefore, the next term in the series is $248$.