Question

find the value of the product

Answer 1

\[\huge{\color{blue}{\prod_{i=0}^a 3^{a+i},a\in R}}\]

Answer 2

\[\huge{\color{blue}{\prod_{i=0}^a 3^{a+i}}}=3^{a}3^{a+1}3^{a+2}3^{a+3}...3^{a+a}\]
using
\[a^ba^c=a^{b+c}\]
\[\large 3^{a}3^{a+1}3^{a+2}3^{a+3}...3^{a+a}=3^{a+a+1+a+2+a+3+...a+a}\]
since
\[a+a+1+a+2+a+3+...a+a=\frac{a+1}{2}(2*a+(a+1)1)\\ \text {we used}S_n=\frac{n}{2}(2a+(n-1)d) \]