Question

determine the value of the sum below.

\\\sum_{k=4}^{10} (-3k + 5)\\

Explanation:

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<reasoning>

Identify the properties of the arithmetic series

\[

$$\begin{aligned} &a_k = -3k + 5\\ &n = 10 - 4 + 1 = 7\\ &a_1 = a_4 = -3(4) + 5 = -7\\ &a_n = a_{10} = -3(10) + 5 = -25 \end{aligned}$$

\]

Apply the arithmetic series sum formula

\[

$$\begin{aligned} &S_n = \frac{n}{2}(a_1 + a_n)\\ &S_7 = \frac{7}{2}(-7 + (-25))\\ &S_7 = \frac{7}{2}(-32)\\ &S_7 = 7 \times (-16) = -112 \end{aligned}$$

\]
</reasoning>

<answer>
\(-112\)
</answer>

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Answer:

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<reasoning>

Identify the properties of the arithmetic series

\[

$$\begin{aligned} &a_k = -3k + 5\\ &n = 10 - 4 + 1 = 7\\ &a_1 = a_4 = -3(4) + 5 = -7\\ &a_n = a_{10} = -3(10) + 5 = -25 \end{aligned}$$

\]

Apply the arithmetic series sum formula

\[

$$\begin{aligned} &S_n = \frac{n}{2}(a_1 + a_n)\\ &S_7 = \frac{7}{2}(-7 + (-25))\\ &S_7 = \frac{7}{2}(-32)\\ &S_7 = 7 \times (-16) = -112 \end{aligned}$$

\]
</reasoning>

<answer>
\(-112\)
</answer>

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