Question

Find the sum of each sequence. \(\sum_{k=1}^{20}(5 k+3)\)

Short answer

1110

Step-by-step solution

Understanding the Sigma Notation

The given expression is \(\textstyle\sum_{k=1}^{20}(5k+3)\), which means we need to find the sum of the expression \(5k+3\) as k ranges from 1 to 20.

Separate the Components

Rewrite the sum as two separate sums for simplification: \(\textstyle\sum_{k=1}^{20}(5k+3)= \sum_{k=1}^{20}5k+\textstyle \sum_{k=1}^{20}3\)

Sum of \(\textstyle \sum_{k=1}^{20}5k\)

Factor out the constant 5: \(5\textstyle\sum_{k=1}^{20}k\). This is the sum of the first 20 natural numbers multiplied by 5. The sum of the first n natural numbers is given by the formula \(\textstyle \sum_{k=1}^{n}k = \frac{n(n+1)}{2}\).

Calculate the Sum of the First 20 Natural Numbers

Using the formula, \(\textstyle \sum_{k=1}^{20}k = \frac{20(20+1)}{2} = 210\). Therefore, \(5\textstyle \sum_{k=1}^{20}k = 5 \times 210 = 1050\).

Sum of \(\textstyle \sum_{k=1}^{20}3\)

Since the summand 3 is a constant, the sum \(3 \times 20 = 60\)

Add the Results

Combine the results from Steps 4 and 5: \(1050 + 60 = 1110\)

Key concepts

Sigma Notation

Sigma notation, represented by the Greek letter \(\textstyle\sum\), stands for the sum of a sequence. In our exercise, we want to find the sum of \(\textstyle\sum_{k=1}^{20}(5k+3)\). It means we need to add every value of the expression \(5k + 3\), starting from \(k = 1\) up to \(k = 20\).

This notation helps simplify writing sequences where you need to add up many terms. Instead of writing out all the terms individually, sigma notation allows us to express large sums concisely. For instance, \(\textstyle\sum_{k=1}^{20} (5k + 3) \) shows we will add the results of \(5 \times 1 + 3\), \(5 \times 2 + 3\), \(5 \times 3 + 3\), and so on, up to \(5 \times 20 + 3\).
Understanding sigma notation will help you handle sum sequences efficiently.

Sum of Natural Numbers

Natural numbers are the numbers we count with: 1, 2, 3, and so forth. Summing natural numbers means adding numbers in this sequence. In our example, we need to sum the first 20 natural numbers as part of solving \(\textstyle\sum_{k=1}^{20} (5k + 3)\).

The sum of the first \(n\) natural numbers is given by the formula:
\(\textstyle\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\).

For \(n = 20\), this formula becomes:
\(\textstyle\sum_{k=1}^{20}k = \frac{20(20+1)}{2} = 210\).

Understanding this formula helps calculate the sum quickly without needing to add each number individually. This concept also lays the foundation for more complex summations.

Summation Formula

The summation formula is a fundamental tool in mathematics, providing a direct way to find the sum of terms in a sequence. For our exercise, we can separate the sum as:

\(\textstyle\sum_{k=1}^{20}(5k + 3) = \textstyle\sum_{k=1}^{20}5k + \textstyle\sum_{k=1}^{20}3\).

We can handle each part separately. First, factor out constants from summation:
\(5\textstyle\sum_{k=1}^{20}k\).

Using the natural numbers sum formula, this part becomes:
\(5 \times 210 = 1050\).

For the constant part, sum constants straightforwardly:
\(3 \times 20 = 60\).

Adding these two results gives the total sum \(1050 + 60 = 1110\).
Summation formulas simplify solving such sequences by breaking them into manageable parts.

Constant Summation

When summing a constant, you repeatedly add the same value a certain number of times. In our exercise, the constant is 3, and it appears 20 times as \(\textstyle\sum_{k=1}^{20}3\).

To understand this, note that adding the same number repeatedly is like multiplying. So for our constant 3 summed 20 times, the result is:
\(3 \times 20 = 60\).

Constant summation simplifies components where values do not change across terms. This type of summation is especially useful in breaking down complex sums. By individually summing constants, we can solve parts of the sequence efficiently before combining results.