+26387 S12 for the series 1 + 1.5 + 2.25 + 3.375 +
\(\begin{array}{|rcll|} \hline && 1 + 1.5 + 2.25 + 3.375 +\ldots \\ &=& 1 + 1.5 + 1.5^2 + 1.5^3 + 1.5^4 + 1.5^5 + \ldots \\ \hline \end{array} \)
\(\begin{array}{|rcll|} \hline A = s_{12} = &=& 1 + 1.5 + 1.5^2 + 1.5^3 + 1.5^4 + 1.5^5 +\ldots + 1.5^{10} + 1.5^{11} \\ B = 1.5\cdot s_{12} = &=& 1.5 + 1.5^2 + 1.5^3 + 1.5^4 + 1.5^5 + 1.5^6 +\ldots + 1.5^{11} + 1.5^{12} \\ B-A = 1.5\cdot s_{12}-s_{12} = &=& 1.5^{12} -1 \\ s_{12}\cdot (1.5-1)&=& 1.5^{12} -1 \\ s_{12}\cdot 0.5&=& 1.5^{12} -1 \\ s_{12} &=& \frac{1.5^{12} -1}{0.5} \\\\ s_{12} &=& \frac{128.746337891}{0.5} \\\\ \mathbf{ s_{12}} &\mathbf{=}& \mathbf{257.492675781\ldots} \\ \hline \end{array} \)
+129838 The common ratio = 1.5....so.....
S12 = [1 - (1.5)^12] / [ 1 - 1.5] = 257.49267578125
+26387 S12 for the series 1 + 1.5 + 2.25 + 3.375 +
\(\begin{array}{|rcll|} \hline && 1 + 1.5 + 2.25 + 3.375 +\ldots \\ &=& 1 + 1.5 + 1.5^2 + 1.5^3 + 1.5^4 + 1.5^5 + \ldots \\ \hline \end{array} \)
\(\begin{array}{|rcll|} \hline A = s_{12} = &=& 1 + 1.5 + 1.5^2 + 1.5^3 + 1.5^4 + 1.5^5 +\ldots + 1.5^{10} + 1.5^{11} \\ B = 1.5\cdot s_{12} = &=& 1.5 + 1.5^2 + 1.5^3 + 1.5^4 + 1.5^5 + 1.5^6 +\ldots + 1.5^{11} + 1.5^{12} \\ B-A = 1.5\cdot s_{12}-s_{12} = &=& 1.5^{12} -1 \\ s_{12}\cdot (1.5-1)&=& 1.5^{12} -1 \\ s_{12}\cdot 0.5&=& 1.5^{12} -1 \\ s_{12} &=& \frac{1.5^{12} -1}{0.5} \\\\ s_{12} &=& \frac{128.746337891}{0.5} \\\\ \mathbf{ s_{12}} &\mathbf{=}& \mathbf{257.492675781\ldots} \\ \hline \end{array} \)
