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Sagot :
Certainly! Let's find the sum of the series [tex]\(\sum_{k=0}^4 (-2k)\)[/tex].
1. Identify the series and the range of summation:
We need to find the sum from [tex]\(k = 0\)[/tex] to [tex]\(k = 4\)[/tex] for the expression [tex]\(-2k\)[/tex]. This means we will calculate:
[tex]\[ (-2 \cdot 0) + (-2 \cdot 1) + (-2 \cdot 2) + (-2 \cdot 3) + (-2 \cdot 4) \][/tex]
2. Calculate each term separately:
[tex]\[ \begin{aligned} &-2 \cdot 0 = 0, \\ &-2 \cdot 1 = -2, \\ &-2 \cdot 2 = -4, \\ &-2 \cdot 3 = -6, \\ &-2 \cdot 4 = -8. \end{aligned} \][/tex]
3. Sum the calculated terms together:
[tex]\[ 0 + (-2) + (-4) + (-6) + (-8). \][/tex]
4. Perform the addition step-by-step:
[tex]\[ \begin{aligned} &0 + (-2) = -2, \\ &-2 + (-4) = -6, \\ &-6 + (-6) = -12, \\ &-12 + (-8) = -20. \end{aligned} \][/tex]
5. Final sum:
[tex]\[ -20 \][/tex]
Therefore, the sum of the series [tex]\(\sum_{k=0}^4 (-2k)\)[/tex] is [tex]\(\boxed{-20}\)[/tex].
1. Identify the series and the range of summation:
We need to find the sum from [tex]\(k = 0\)[/tex] to [tex]\(k = 4\)[/tex] for the expression [tex]\(-2k\)[/tex]. This means we will calculate:
[tex]\[ (-2 \cdot 0) + (-2 \cdot 1) + (-2 \cdot 2) + (-2 \cdot 3) + (-2 \cdot 4) \][/tex]
2. Calculate each term separately:
[tex]\[ \begin{aligned} &-2 \cdot 0 = 0, \\ &-2 \cdot 1 = -2, \\ &-2 \cdot 2 = -4, \\ &-2 \cdot 3 = -6, \\ &-2 \cdot 4 = -8. \end{aligned} \][/tex]
3. Sum the calculated terms together:
[tex]\[ 0 + (-2) + (-4) + (-6) + (-8). \][/tex]
4. Perform the addition step-by-step:
[tex]\[ \begin{aligned} &0 + (-2) = -2, \\ &-2 + (-4) = -6, \\ &-6 + (-6) = -12, \\ &-12 + (-8) = -20. \end{aligned} \][/tex]
5. Final sum:
[tex]\[ -20 \][/tex]
Therefore, the sum of the series [tex]\(\sum_{k=0}^4 (-2k)\)[/tex] is [tex]\(\boxed{-20}\)[/tex].
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