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\includegraphics[max width=\textwidth, alt={}, center]{1b9c6b41-69dd-4132-92c7-9507cbd741dd-08_542_661_269_731} The diagram shows the curve \(y = 3 \mathrm { e } ^ { 2 x - 1 }\). The shaded region is bounded by the curve and the lines \(x = a , x = a + 1\) and \(y = 0\), where \(a\) is a constant. It is given that the area of the shaded region is 120 square units.

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 80 + \mathrm { e } ^ { 2 a - 1 } \right) - \frac { 1 } { 2 }\).
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.