Challenging Indeterminate Series Pattern
The expression provided is:
\[ \frac{5}{2^3} + 3 \cdot 2^1 + \frac{7}{2^5} + 4 \cdot 2^6 \]
To put this expression in sigma notation, we want to find a pattern that represents all terms in the series.
To do this, let's look at each term and its index and see if we can identify a pattern:
1. For the first term, the index is 3, and the coefficient is 5, but since it's divided by \(2^3\), we must flip the sign in the exponent to get the exact term.
2. For the second term, the index is 1, and the coefficient is 3 times 2 raised to the power of the index.
3. For the third term, the index is 5, and the coefficient is 7, divided by \(2^5\).
4. For the fourth term, the index is 6, and the coefficient is 4 times 2 raised to the power of the index.
We want a general formula to describe this sequence. Notice that the even indices have coefficients 3 and 4, which are multiplied by the base 2 raised to a positive power, while the odd indices have coefficients 5 and 7, which are seemingly unrelated and divided by the base 2 raised to the power of the index.
There is no single pattern encompassing all the terms given with a simple arithmetic progression in the exponents or coefficients, and none of the sigma notations provided seem to accurately represent the given expression in a correct and patterned form. They each have k as an index, but the terms in the provided expression do not follow a pattern that could be encapsulated by k running from a beginning to an end value. Hence, there may be an error in the problem or the provided sigma notations, because they do not correspond to the given expression.
