3.1 Consider the quadratic number pattern: \( 3 ; 7 ; 12 ; \ldots \) Show that the general term of this number patten is given by \( T_{n}=\frac{1}{2} n^{2}+\frac{5}{2} \) ?

To derive the general term \( T_n = \frac{1}{2} n^2 + \frac{5}{2} \) for the given pattern, we first observe the differences between consecutive terms: \( 7 - 3 = 4 \) and \( 12 - 7 = 5 \). The first differences are \( 4 \) and \( 5 \), suggesting the second differences are constant (which equals \( 1 \)). This is indicative of a quadratic sequence. Assuming a quadratic formula \( T_n = an^2 + bn + c \), we can set up a system using the initial conditions: when \( n=1, T_1=3 \); when \( n=2, T_2=7 \); and when \( n=3, T_3=12 \). Solving this system provides the coefficients \( a = \frac{1}{2}, b = 0, c = \frac{5}{2} \). Therefore, we can confirm that the general term is indeed \( T_n = \frac{1}{2} n^2 + \frac{5}{2} \). To visualize this quadratic progression, imagine plotting the points on a graph! The term values \( (1,3), (2,7), (3,12) \) form a parabolic curve. That classic upward curvature is a telltale sign of quadratics in the mathematical world, revealing how they grow faster than linear sequences. So, next time you see a pattern like this, think of it as a fun curve waiting to be explored! Also, if you're diving deeper into quadratic functions, consider getting into polynomial growth rates. They're not just for textbooks; these forms are sprinkled throughout nature, economics, and engineering! From predicting rocket trajectories to calculating maximum profit in business models, understanding quadratics can turn you into a mathematical wizard in real-world scenarios.