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Q 79.

Expert-verified
Found in: Page 809

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 71-82, find the sum of each sequence.$\sum _{k=10}^{60}\left(2k\right)$

The sum of this sequence is 3570.

See the 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:

$\underset{k=1}{\overset{n}{\sum \left(c{a}_{k}\right)}}=c\underset{k=1}{\overset{n}{\sum {a}_{k}}}\phantom{\rule{0ex}{0ex}}So,\phantom{\rule{0ex}{0ex}}\underset{k=1}{\overset{n}{\sum \left(2k\right)}}=2\underset{k=1}{\overset{n}{\sum k}}$

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

The formula for summation:

$\underset{k=10}{\overset{60}{\sum k}}=\underset{k=1}{\overset{60}{\sum k}}\underset{k=1}{\overset{9}{-\sum k}}\phantom{\rule{0ex}{0ex}}⇒\underset{k=10}{\overset{60}{\sum k}}=\frac{60\left(60+1\right)}{2}-\frac{9\left(9+1\right)}{2}\phantom{\rule{0ex}{0ex}}⇒\underset{k=10}{\overset{60}{\sum k}}=30×61-9×5\phantom{\rule{0ex}{0ex}}⇒\underset{k=10}{\overset{60}{\sum k}}=1830-45\phantom{\rule{0ex}{0ex}}⇒\underset{k=10}{\overset{60}{\sum k}}=1785$

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

Using the property of sequences gives:

$\underset{k=10}{\overset{60}{\sum k}}=1785\phantom{\rule{0ex}{0ex}}⇒\underset{k=10}{\overset{60}{2\sum k}}=2×1785\phantom{\rule{0ex}{0ex}}⇒\underset{k=10}{\overset{60}{2\sum k}}=3570$