           10           10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
           k = 1          k = 1

  10
c. Σ (aₖ + bₖ - 1)
  k = 1

Guida verificata passo dopo passo

Identify the given sums: \( \sum_{k=1}^{10} a_k = -2 \) and \( \sum_{k=1}^{10} b_k = 25 \).

Write the expression to find: \( \sum_{k=1}^{10} (a_k + b_k - 1) \).

Use the property of summation that allows splitting sums: \( \sum_{k=1}^{10} (a_k + b_k - 1) = \sum_{k=1}^{10} a_k + \sum_{k=1}^{10} b_k - \sum_{k=1}^{10} 1 \).

Substitute the known sums into the expression: \( = (-2) + 25 - \sum_{k=1}^{10} 1 \).

Calculate \( \sum_{k=1}^{10} 1 \) as the sum of ten ones, which equals 10, then combine all terms to express the final sum.

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Properties of Summation

Summation properties allow us to break down or combine sums term-by-term. For example, the sum of a sum is the sum of the sums: Σ(aₖ + bₖ) = Σaₖ + Σbₖ. This linearity helps simplify complex summations by separating or combining known series.

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Properties of Functions

Constant Term in Summation

When summing a constant over n terms, the result is the constant multiplied by n: Σc = n * c. This is useful when the summation includes a constant term added or subtracted from each element, allowing easy calculation of that part of the sum.

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Divergence Test (nth Term Test)

Given Summation Values

Knowing the values of Σaₖ and Σbₖ provides a foundation to find related sums. By substituting these known sums into expressions involving aₖ and bₖ, we can compute new summations without evaluating each term individually.

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