Pregunta
Evaluate the summation below. \[ 3 \sum_{d=0}^{3}(-d-1) \]
Ask by Blake Carrillo. in the United States
Mar 10,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The summation evaluates to -30.
Solución
We are given the expression
\[
3 \sum_{d=0}^{3} \left(-d-1\right).
\]
**Step 1. Evaluate the summation.**
For each \( d \) from \( 0 \) to \( 3 \), compute \(-d-1\):
- For \( d=0 \): \(-0-1=-1\)
- For \( d=1 \): \(-1-1=-2\)
- For \( d=2 \): \(-2-1=-3\)
- For \( d=3 \): \(-3-1=-4\)
Thus, the summation becomes:
\[
\sum_{d=0}^{3} \left(-d-1\right) = (-1) + (-2) + (-3) + (-4).
\]
Compute the sum:
\[
-1 -2 -3 -4 = -10.
\]
**Step 2. Multiply by 3.**
Now multiply the sum by \( 3 \):
\[
3 \times (-10) = -30.
\]
The final result is:
\[
-30.
\]
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución The Deep Dive
To evaluate the summation \[ 3 \sum_{d=0}^{3}(-d-1), \] we first calculate the inner summation \( \sum_{d=0}^{3}(-d-1) \). We can break this down by calculating \(-d-1\) for each value of \(d\) from 0 to 3: - For \(d = 0\): \[ -0 - 1 = -1. \] - For \(d = 1\): \[ -1 - 1 = -2. \] - For \(d = 2\): \[ -2 - 1 = -3. \] - For \(d = 3\): \[ -3 - 1 = -4. \] Now, we can compute the sum: \[ \sum_{d=0}^{3}(-d-1) = -1 + (-2) + (-3) + (-4). \] Calculating this gives: \[ -1 - 2 - 3 - 4 = -10. \] Next, we multiply this result by 3: \[ 3 \sum_{d=0}^{3}(-d-1) = 3 \times (-10) = -30. \] Thus, the value of the original summation is \[ \boxed{-30}. \]
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