           20           20
Suppose that Σ aₖ = 0 and Σ bₖ = 7. Find the value of
           k = 1          k = 1

  20
a. Σ 3aₖ
  k = 1

Guida verificata passo dopo passo

Identify the given information: \( \sum_{k=1}^{20} a_k = 0 \) and \( \sum_{k=1}^{20} b_k = 7 \).

Focus on the expression to find: \( \sum_{k=1}^{20} 3a_k \). Notice that the summation is over \( 3a_k \), which means each term \( a_k \) is multiplied by 3.

Use the property of summations that allows constants to be factored out: \( \sum_{k=1}^{n} c \cdot x_k = c \sum_{k=1}^{n} x_k \). Applying this, we get \( \sum_{k=1}^{20} 3a_k = 3 \sum_{k=1}^{20} a_k \).

Substitute the known value of \( \sum_{k=1}^{20} a_k = 0 \) into the expression: \( 3 \times 0 \).

Conclude that the value of \( \sum_{k=1}^{20} 3a_k \) is \( 3 \times 0 \), which simplifies to 0.

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Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms indexed by an integer variable. Understanding how to interpret and manipulate these sums is essential for evaluating expressions involving series, such as Σ aₖ or Σ 3aₖ.

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04:22

Sigma Notation

Properties of Summations

Summations have linearity properties, meaning constants can be factored out and sums can be split or combined. For example, Σ c·aₖ = c·Σ aₖ, which allows simplification of expressions like Σ 3aₖ by factoring out the constant 3.

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Properties of Functions

Given Series Values and Their Use

Knowing the values of given sums, such as Σ aₖ = 0 and Σ bₖ = 7, is crucial for solving problems. These values can be substituted directly into expressions to find unknown sums or simplify calculations.

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