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Chapter 11: Problem 33

Find the sum. $$\sum_{k=1}^{40}(2 k+3)$$

Short Answer

Expert verified
The sum is 1760.

Step by step solution

01

- Understand the Sum Notation

The notation \(\sum_{k=1}^{40}(2k+3)\) means that we need to sum the expression \(2k+3\) for each integer value of \(k\) from \(1\) to \(40\).
02

- Split the Sum

The sum \(\sum_{k=1}^{40}(2k+3)\) can be split into two separate sums: \(\sum_{k=1}^{40}2k + \sum_{k=1}^{40}3\). This uses the properties of sums to separate constant multiples and additive terms.
03

- Sum Each Part Separately

First, consider \(\sum_{k=1}^{40} 2k\). This can be factored as \(2 \sum_{k=1}^{40} k\). We know that the sum of the first \(n\) natural numbers is given by the formula \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\). Here, \(n = 40\), so \(\sum_{k=1}^{40} k = \frac{40 \cdot 41}{2} = 820\). Therefore, \(2 \sum_{k=1}^{40} k = 2 \cdot 820 = 1640\).
04

- Sum the Constant

Next, consider \(\sum_{k=1}^{40}3\). Since \(3\) is a constant added for each of the \(40\) values of \(k\), this sum is simply \(3 \cdot 40 = 120\).
05

- Combine the Results

Add the results of the two separate sums: \(1640 + 120 = 1760\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

summation notation
Summation notation is a way to represent the sum of a sequence of terms. It's vital for understanding many mathematical concepts.
The symbol \(\sum\) (Sigma) is used to denote summation. The expression \(\sum_{k=1}^{40} (2k + 3)\) tells us to sum the terms generated by the expression \(2k + 3\) as k takes every integer value from 1 to 40.
This compact notation saves space and makes formulas easier to read and understand.
natural numbers sum formula
The natural numbers sum formula helps us quickly find the sum of the first 'n' natural numbers without adding each number individually.
The formula is given by: \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\).
This formula lets us find the sum of natural numbers up to any 'n'. For example, to sum the first 40 numbers, we substitute 'n' with 40:
\(\frac{40 \cdot 41}{2} = 820\).

Using this, we can easily find sums involving natural numbers without lengthy calculations.
separating sums
Separating sums is a technique where we break a complex summation into simpler parts.
This makes it easier to solve. For instance, \(\sum_{k=1}^{40} (2k + 3)\) can be split into two separate sums: \(\frac{2 \sum_{k=1}^{40} k\rightarrow\) and \(\frac{3 \sum_{k=1}^{40} 1\) as separate sums.
This uses the distributive property of sums, allowing us to handle each part independently before combining them.

This method simplifies the solving process, making complex problems more manageable.

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