Chapter 9 - Section 9.1 - Sequences - 9.1 Assess Your Understanding - Page 647: 49

$$\sum_{k=1}^{n}(k+2) = \dfrac{n(n+1)}{2} + 2n$$

Work Step by Step

RECALL: For any constant $c$, (1) $$\sum_{k=1}^{n}(k+c) = \sum_{k=1}^{n}k + \sum_{k=1}^{n}c$$ (2) $$\sum_{i=1}^{n}c = nc$$ (3) $$\sum_{k=1}^{n}k = \dfrac{n(n+1)}{2}$$ Using rule (1) above gives: $$\sum_{k=1}^n(k + 2) = \sum_{k=1}^{n}k + \sum_{k=1}^{n}2$$ Use rule (3) and rule (2) above, respectively, to obtain: $$\sum_{k=1}^{n}k + \sum_{k=1}^{n}2 = \dfrac{n(n+1)}{2} + 2n$$ Thus, $$\sum_{k=1}^{n}(k+2) = \dfrac{n(n+1)}{2} + 2n$$

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