Sequences, Rules and Sums

It might be easier to write the general term like

$\begin{aligned} a_n &= a_{n-1}(1+\cfrac{3}{a_{n-1}}) = a_{n-1} + 3 \hspace{5mm} \color{#3D99F6} a_n > 1 \text{ as } a_1 \text{ and } (1+\cfrac{3}{a_{n-1}}) > 1 \\ \{a_n\} &= {3, 6, 9, ....} \\ a_1 + ...+ a_{100} &= 3\times(1+2+...+100) = 15150 \end{aligned}$

Let us call any arbitrary term $x$ . Let the next term of $x$ be $y$ . Then, from the rule, consecutive multiplication factor from $x$ to $y$ (i.e. the ratio of the next term $y$ to this term $x$ ) will be $1+3×\frac{1}{x}=1+\frac{3}{x}$ . But this ratio is also equivalent to $\frac{y}{x}$ . Therefore, $\frac{y}{x}=1+\frac{3}{x}$ . Multiplying both sides by $x$ , $x×\frac{y}{x}=x+x×\frac{3}{x}$ . Now let us carry out induction. Base Case: The first term (namely $3$ ) is positive. Induction step: If $x$ is positive, then in the previous equation, we can safely cancel the $x$ in the numerator and the $x$ in the denominator, in the left-hand side, and also cancel out $x$ in the numerator and the $x$ in the denominator, in the second term of the right-hand side, which leads to $y=x+3$ , which shows that a positive $3$ added to a positive $x$ leads a positive $y$ . Therefore, by induction we have proved that all terms are positive (and therefore non-zero). Therefore, the consequent of induction step, is also true, implying that $y=x+3$ , where $x$ was any arbitrary term, where $y$ was the next term of $x$ , implies that, any term except for the first one, is the previous term plus $3$ . Therefore the 100th term will be the first term plus ninety-nine threes, which is equal to $3+99×3=100×3=300$ . Therefore, the sum of the first 300 terms is equal to $3+6+9+12+\dots 291+294+297+300=3×(1+2+3+4+\dots 97+98+99+100)$ . Using the $n$ th Triangular Number Formula, a special case of the $n$ th term of an arithmetic series formula, $1+2+3+4+\dots 97+98+99+100=\frac{1}{2}×101×100$ . Substituting this for $1+2+3+4+\dots 97+98+99+100$ in this expression: $3×(1+2+3+4+\dots 97+98+99+100) = \frac{1}{2}×3×101×100=15150$ .