Let \( T_{n}=4 n^{2}+n \) be the general term of a quadratic sequence with the fourth term being 68. 21 and 29 are the second and third terms, respectively, of the sequence of the first differences of the quadratic sequence.

The general term of a quadratic sequence is given by \( T_n = 4n^2 + n \). To find the fourth term, we can substitute \( n=4 \): \[ T_4 = 4(4^2) + 4 = 4(16) + 4 = 64 + 4 = 68 \] This confirms the fourth term is indeed 68. Now, the first differences of the quadratic sequence are calculated by finding \( T_{n+1} - T_n \). Expressing this calculation gives us a linear sequence since the first differences of quadratic sequences are linear. The first term is given by \( T_1 = 4(1^2) + 1 = 5 \). The second term is \( T_2 = 4(2^2) + 2 = 18 \). The third term is \( T_3 = 4(3^2) + 3 = 39 \). The first differences are \( T_2 - T_1 = 18 - 5 = 13 \) (first difference), \( T_3 - T_2 = 39 - 18 = 21 \) (second difference), and \( T_4 - T_3 = 68 - 39 = 29 \) (third difference). This leads to the first differences being \( 13, 21, 29, ...\). The pattern in the first differences helps to reveal that each difference is increasing by 8, consistent with its quadratic nature. In terms of historical context, quadratic sequences have fascinated mathematicians since ancient times. Notably, they relate to polynomial growth, which is foundational not only in theoretical mathematics but also in areas like physics and economics, where relationships are often modeled using quadratic equations. For practical applications, understanding quadratic sequences can help decipher various real-world problems. For instance, in projectile motion, the height of an object at a specific time can be modeled as a quadratic equation. Knowing how to derive and work with these sequences allows for predictions about the object's behavior at various time intervals, assisting engineers and physicists in designs and calculations.