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Q 80.

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.$\sum _{k=8}^{40}\left(-3k\right)$

The sum of this sequence is -2376.

See the step by step solution

## Step 1. Write the given information.

The sum of sequence:

$\sum _{k=8}^{40}\left(-3k\right)$

## Step 2. Use the property of sequences.

Using the property of sequences:$\underset{k=1}{\overset{n}{\sum \left(c{a}_{k}\right)}}=c\underset{k=1}{\overset{n}{\sum {a}_{k}}}\phantom{\rule{0ex}{0ex}}So,\phantom{\rule{0ex}{0ex}}\underset{k=1}{\overset{n}{\sum \left(-3k\right)}}=-3\underset{k=1}{\overset{n}{\sum k}}$

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

$\underset{k=8}{\overset{40}{\sum k}}=\underset{k=1}{\overset{40}{\sum k}}\underset{k=1}{\overset{7}{-\sum k}}\phantom{\rule{0ex}{0ex}}⇒\underset{k=8}{\overset{40}{\sum k}}=\frac{40\left(40+1\right)}{2}-\frac{7\left(7+1\right)}{2}\phantom{\rule{0ex}{0ex}}⇒\underset{k=8}{\overset{40}{\sum k}}=20×41-7×4\phantom{\rule{0ex}{0ex}}⇒\underset{k=8}{\overset{40}{\sum k}}=820-28\phantom{\rule{0ex}{0ex}}⇒\underset{k=8}{\overset{40}{\sum k}}=792$

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

Using the property of sequences gives:

$\underset{k=8}{\overset{40}{\sum k}}=792\phantom{\rule{0ex}{0ex}}⇒\left(-\underset{k=8}{\overset{40}{3\right)\sum k}}=-3×792\phantom{\rule{0ex}{0ex}}⇒\underset{k=8}{\overset{40}{\left(-3\right)\sum k}}=-2376$