The correct answer is C) $127$. To find the next term in the series, we will analyze the pattern through two distinct methods: analysis of successive differences and direct formula recognition. **Method 1: Analysis of Successive Differences** 1. **Calculate the first-order differences** (differences between consecutive terms): $10 - 3 = 7$ $29 - 10 = 19$ $66 - 29 = 37$ The sequence of first differences is: $7, 19, 37$ 2. **Calculate the second-order differences** (differences between the first-order differences): $19 - 7 = 12$ $37 - 19 = 18$ The sequence of second differences is: $12, 18$ 3. **Calculate the third-order differences** (differences between the second-order differences): $18 - 12 = 6$ Here, we observe that the third-order difference is a constant value of $6$. 4. **Extend the pattern to find the next terms in the difference sequence:** * Assuming the third-order difference remains constant at $6$, the next second-order difference will be $18 + 6 = 24$. * The next first-order difference will be the last observed first-order difference plus the projected second-order difference: $37 + 24 = 61$. * The next term in the main series will be the last given term plus the projected first-order difference: $66 + 61 = 127$. **Method 2: Pattern Recognition (Direct Formula)** Upon closer inspection, each term in the series can be expressed as a perfect cube of its position number plus a constant value. Let $T_n$ denote the $n$-th term in the series: * For $n=1$: $T_1 = 3 = 1^3 + 2 = 1 + 2$ * For $n=2$: $T_2 = 10 = 2^3 + 2 = 8 + 2$ * For $n=3$: $T_3 = 29 = 3^3 + 2 = 27 + 2$ * For $n=4$: $T_4 = 66 = 4^3 + 2 = 64 + 2$ Following this established pattern, the next term, $T_5$, would be: * For $n=5$: $T_5 = 5^3 + 2 = 125 + 2 = 127$ Both analytical methods consistently lead to the same result. The series follows the pattern $T_n = n^3 + 2$. Therefore, the next term in the series is $127$.