Question

In Problems 71-82, find the sum of each sequence.

$\sum _{k=10}^{60}\left(2k\right)$

Short answer

The sum of this sequence is 3570.

Step-by-step solution

Step 1. Write the given information. 

The sum of sequence:

$\sum _{k=10}^{60}\left(2k\right)$

Step 2. Use the property of sequences.

Using the property of sequences:

$$\begin{gathered} \underset{k=1}{\overset{n}{\sum \left(c{a}_k\right)}}=c\underset{k=1}{\overset{n}{\sum a_k}} \\ So, \\ \underset{k=1}{\overset{n}{\sum \left(2k\right)}}=2\underset{k=1}{\overset{n}{\sum k}} \end{gathered}$$

Step 3. Use the formula for sum of sequences of n real numbers.

The formula for summation:

$$\begin{gathered} \underset{k=10}{\overset{60}{\sum k}}=\underset{k=1}{\overset{60}{\sum k}}\underset{k=1}{\overset{9}{-\sum k}} \\ \Rightarrow \underset{k=10}{\overset{60}{\sum k}}=\frac{60(60+1)}{2}-\frac{9(9+1)}{2} \\ \Rightarrow \underset{k=10}{\overset{60}{\sum k}}=30\times 61-9\times 5 \\ \Rightarrow \underset{k=10}{\overset{60}{\sum k}}=1830-45 \\ \Rightarrow \underset{k=10}{\overset{60}{\sum k}}=1785 \end{gathered}$$

Step 4. Use the property of sequences from Step 2.

Using the property of sequences gives:

$$\begin{gathered} \underset{k=10}{\overset{60}{\sum k}}=1785 \\ \Rightarrow \underset{k=10}{\overset{60}{2\sum k}}=2\times 1785 \\ \Rightarrow \underset{k=10}{\overset{60}{2\sum k}}=3570 \end{gathered}$$