Explanation Given Information: The provided sum is ∑ i = 1 n ( 1 − 2 i n + i 2 n 2 ) ( 3 n ) . Formula used: The value of the sum ∑ k = 1 n ( x k ± y k ) = ∑ k = 1 n x k ± ∑ k = 1 n y k . The value of the sum ∑ k = 1 n k = n ( n + 1 ) 2 . The value of the sum ∑ k = 1 n 1 = n . The value of the sum ∑ k = 1 n c x k = c ∑ k = 1 n x k . The value of the sum ∑ k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6 . Calculation: Consider the sum ∑ i = 1 n ( 1 − 2 i n + i 2 n 2 ) ( 3 n ) . This can be written as, ∑ i = 1 n ( 1 − 2 i n + i 2 n 2 ) ( 3 n ) = ∑ i = 1 n ( 3 n − 3 n ⋅ 2 i n + 3 n ⋅ i 2 n 2 ) = ∑ i = 1 n ( 3 n − 6 i n 2 + 3 i 2 n 3 ) Recall that the value of the sum ∑ k = 1 n ( x k ± y k ) = ∑ k = 1 n x k ± ∑ k = 1 n y k . Thus, ∑ i = 1 n ( 1 − 2 i n + i 2 n 2 ) ( 3 n ) = ∑ i = 1 n ( 3 n − 6 i n 2 + 3 i 2 n 3 ) = ∑ i = 1 n 3 n − ∑ i = 1 n ( 6 i n 2 ) + ∑ i = 1 n ( 3 i 2 n 3 ) Recall that the value of the sum ∑ k = 1 n c x k = c ∑ k = 1 n x k . Thus, ∑ i = 1 n ( 1 − 2 i n + i 2 n 2 ) ( 3 n ) = ∑ i = 1 n 3 n − ∑ i = 1 n ( 6 i n 2 ) + ∑ i = 1 n ( 3 i 2 n 3 ) = 3 n ∑ i = 1 n 1 − 6 n 2 ∑ i = 1 n i + 3 n 3 ∑ i = 1 n i 2

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ISBN: 9780357865095

Author: HARSHBARGER

Publisher: CENGAGE L

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Counting Principles

In statistics, the counting principle can be said to be the concept in which the quantity of an item is determined. There can be a single item, or a group of items, or items that are related to each other. In every possible set of items, the quantity of …

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Chapter 13.1, Problem 25E

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To calculate: The expression of ∑i=1n(1−2in+i2n2)(3n) without the summation symbol.

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Chapter 13.1, Problem 24E

Chapter 13.1, Problem 26E

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Ch. 13.1 - CHECKPOINT 1. For the interval [0, 2]. determine...Ch. 13.1 - Prob. 2CP Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E

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The expression of ∑ i = 1 n ( 1 − 2 i n + i 2 n 2 ) ( 3 n ) without the summation symbol.

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To calculate: The expression of ∑i=1n(1−2in+i2n2)(3n) without the summation symbol.

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Chapter 13 Solutions