Suppose that ∑k=1^n ak=0 and ∑k=1^n bk=1 . Find the values of a. ∑k=1^n 8 ak b. ∑k=1^n 2

step 1 In question number 18, in section 1, we have that submission for A of K from K equal 1 to n equa

Suppose that ∑k=1^n ak=0 and ∑k=1^n bk=1 . Find the values...

Suppose that $\sum_{k=1}^{n} a_{k}=0$ and $\sum_{k=1}^{n} b_{k}=1 .$ Find the values of $$ \begin{array}{ll}{\text { a. } \sum_{k=1}^{n} 8 a_{k}} & {\text { b. } \sum_{k=1}^{n} 250 b_{k}} \\ {\text { c. } \sum_{k=1}^{n}\left(a_{k}+1\right)} & {\text { d. } \sum_{k=1}^{n}\left(b_{k}-1\right)}\end{array} $$

Suppose that $\sum_{k=1}^{n} a_{k}=0$ and $\sum_{k=1}^{n} b_{k}=1 .$ Find the values of
$$
\begin{array}{ll}{\text { a. } \sum_{k=1}^{n} 8 a_{k}} & {\text { b. } \sum_{k=1}^{n} 250 b_{k}} \\ {\text { c. } \sum_{k=1}^{n}\left(a_{k}+1\right)} & {\text { d. } \sum_{k=1}^{n}\left(b_{k}-1\right)}\end{array}
$$

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Key Concepts

Linearity of Summation

This concept refers to the property that the sum of a linear combination of sequences is equal to the corresponding linear combination of their sums. It allows breaking up sums of terms that involve additions and multiplications by constants, thereby simplifying the computation of complex sums by handling each part separately.

Scalar Multiplication in Summations

Scalar multiplication in summations states that a constant multiplied by each term in a sum can be factored out of the summation. This property is essential for handling problems where each term in a sequence is multiplied by the same constant, such as computing the sum 8a?, which is equivalent to 8 times the sum of a?.

Summation of Constant Terms

Summing constant values over a range is straightforward: it equals the constant multiplied by the number of terms being summed. This concept is useful when constants are added to or subtracted from a variable in a summation, simplifying the evaluation process by converting the sum into a multiplication problem.

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