Express the sum of the first 50 terms of the sequence in sigma notation: −1,−16,−136, . . .

Question

.(1 point)
Responses

∑50n=1(−16)(−1)n−1
sigma-summation underscript n equals 1 overscript 50 End Scripts left parenthesis negative Start Fraction 1 over 6 End Fraction right parenthesis left parenthesis negative 1 right parenthesis superscript n minus 1 baseline

∑50n=1(−1)(16)n−1
sigma-summation underscript n equals 1 overscript 50 End Scripts left parenthesis negative 1 right parenthesis left parenthesis Start Fraction 1 over 6 End Fraction right parenthesis superscript n minus 1 baseline

∑50n=1(−1)(50)n−1
sigma-summation underscript n equals 1 overscript 50 End Scripts left parenthesis negative 1 right parenthesis left parenthesis 50 right parenthesis superscript n minus 1 baseline

∑1n=50(−1)(−16)n−1

Answer 1

To express the sum of the first 50 terms of the sequence \(-1, -16, -136, \ldots\) in sigma notation, we first need to observe the pattern of the sequence.

1. Identifying the pattern:

   • The first term is \(-1\) (which can be seen as \(-1 \cdot 1\)).
   • The second term is \(-16\) (which can be expressed as \(-1 \cdot 16\)).
   • The third term is \(-136\) (which can be expressed as \(-1 \cdot 136\)).

2. Finding the general term:

   • We notice that the absolute values of the terms appear to follow a particular pattern of multiplying by a factor related to 16: \[ 16, 136, \ldots \]
   • This suggests exponential growth. However, it doesn't seem to follow a clear geometric or arithmetic series neatly.

3. Conjecturing a general term formula: Upon examining the terms, realizing they can be expressed as: \[ a_n = -1 \cdot c_n \] where \(c_n\) represents some pattern. Since further inspection would be needed to analyze \(c_n\), we can provide a related specific format for sigma notation.

4. Final Formulation: We can express each term using a generated general term format:

   • If we consider that each term takes the form:
   • It seems to generate a sequence potentially with \((−1)^{n}\) modifiers or similar adjustable constructs.

From the given options, the closest expression that matches our suspected construction of \(a_n\) based on variable sequences and transformations would then be:

\[ \sum_{n=1}^{50} (-16)(-1)^{n-1} \]

This option brings clear compatibility with existing sequences and translations. Thus, without the precise form used in expressing the transformation of initial terms drastically leading to conclusions, this representation could favorably be selected.

So, the final expression as part of the sigma notation that matches the terms of the sequence is appropriately:

\[ \sum_{n=1}^{50} (-16)(-1)^{n-1} \]

This indicates the alternating sign with subsequent angular liquescence for a mathematically valid representation of the first 50 terms of the structure in question.