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Chapter 4: Problem 32

Use the rules of summation and the summation formulas to evaluate the sum. $$ \sum_{k=1}^{10} k(k-2) $$

Short Answer

Expert verified
The sum of the given expression \(\sum_{k=1}^{10} k(k-2)\) is evaluated as follows: first, apply the properties of summation and separate the sum into two individual sums \(\sum_{k=1}^{10} k^2 - \sum_{k=1}^{10} 2k\). Then, use summation formulas and properties to find the sum of each part. Finally, subtract both results to obtain the final sum: \(\sum_{k=1}^{10} k(k-2) = 165\).

Step by step solution

01

Apply the properties of summation

Use the properties of summation to split the sum into two simpler sums: \[ \sum_{k=1}^{10} k(k-2) = \sum_{k=1}^{10} (k^2 - 2k) \]
02

Separate the two sums

As summation obeys distributive property, we can separate the given sum into two individual sums: \[ \sum_{k=1}^{10} (k^2 - 2k) = \sum_{k=1}^{10} k^2 - \sum_{k=1}^{10} 2k \]
03

Apply the summation formulas

Use the summation formulas for the first sum and the properties of summation for the second sum: \[ \sum_{k=1}^{10} k^2 = \frac{n(n+1)(2n+1)}{6} = \frac{10(10+1)(2(10)+1)}{6} = \frac{10 \cdot 11 \cdot 21}{6} \] \[ \sum_{k=1}^{10} 2k = 2 \sum_{k=1}^{10} k = 2 \frac{n(n+1)}{2} = 2 \frac{10(10+1)}{2} = 2 \cdot 10 \cdot 11 \]
04

Evaluate the sums

Now, compute both sums and subtract them to get the final result: \[ \sum_{k=1}^{10} k(k-2) = \frac{10 \cdot 11 \cdot 21}{6} - 2 \cdot 10 \cdot 11 = 385-220 = 165 \] The sum of the given expression from k=1 to k=10 is 165.

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