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Answer :
Final answer:
The sum 1 + 5 + 25 + 125 + 625 + 3125 written in sigma notation is \(\sum_{n=1}^{6} 5^{n-1}\).
Explanation:
The given series is a geometric progression with each term being $5$ times the previous term, starting with $1$. To write this sum in sigma notation, observe that each term can be expressed as $5^{n-1}$, where $n$ is the position of the term in the sequence. Thus, the first term is $5^{0}$ (since $5^{0} = 1$), the second term is $5^{1}$, and so on, up to the sixth term which is $5^{5}$. The sigma notation for the given series is:
\[\sum_{n=1}^{6} 5^{n-1}\]
This notation sums up the expression $5^{n-1}$ as $n$ goes from $1$ to $6$.
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