           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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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