Answer:
[tex]\textbf{The value of the first term $a = 24$ and the common difference $r = 2$}\\[/tex]
Step-by-step explanation:
[tex]\textup{The general form of the GP would be:}\\$a, ar, ar^2, ar^3, . . .$$\therefore $The $n^{th}$ term would be $ar^{n-1}$\\Also, it is stated that the sixth term of the GP is $768$\\ \[768 = ar^5 \tag{1}\] \\\textup{Now consider the second GP with the first term $a$ and with common difference $6r$}\\[/tex]
[tex]\textup{Therefore, the GP would be:}\\$a, a(6r), a{(6r)}^2, a{(6r)}^3,. . .$\\\textup{Consequently the $n^{th}$ term would be:} $a{{(6r)}^{n-1}}$\\Now given its third term is $3456$\\$\implies 3456 = a{(6r)}^2\\$ \[96 = ar^2 \tag{2}\] $ \frac{\textcircled{1}}{\textcircled{2}} \implies \frac{ar^5}{ar^2} $ = $\frac{768}{96}$\\ $\implies r^3 = 8$\therefore r = 2$\\\textup{Now substituting $r=2$ in $(2)$, we get $a = 24.$ }[/tex]
