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Chapter 14: Problem 61

Find each sum. $$ \sum_{k=1}^{4}(6 k) $$

Short Answer

Expert verified
The sum is 60.

Step by step solution

01

Understand the Summation Formula

The sum \( \sum_{k=1}^{4}(6k) \) involves summing the values of the expression \( 6k \) as \( k \) takes each integer value from 1 to 4.
02

Substitute and Calculate for Each Term

Substitute \( k = 1, 2, 3, 4 \) into the expression \( 6k \):- For \( k = 1 \), \( 6 \times 1 = 6 \).- For \( k = 2 \), \( 6 \times 2 = 12 \).- For \( k = 3 \), \( 6 \times 3 = 18 \).- For \( k = 4 \), \( 6 \times 4 = 24 \).
03

Sum the Calculated Terms

Add the results from each evaluation: \( 6 + 12 + 18 + 24 \).
04

Calculate the Final Sum

Perform the addition:\( 6 + 12 = 18 \),\( 18 + 18 = 36 \),\( 36 + 24 = 60 \).The final sum is 60.

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Key Concepts

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

Understanding Arithmetic Sequences
Arithmetic sequences are lists of numbers where each number after the first is a constant addition to the previous one. This constant is known as the common difference. For example, in the sequence 3, 6, 9, 12, the common difference is 3 because each number increases by 3 from the previous number.
To determine any term in an arithmetic sequence, simply use the formula:- \(a_n = a_1 + (n-1) \cdot d\)Where:- \(a_n\) is the \(n\)-th term,- \(a_1\) is the first term,- \(d\) is the common difference, and- \(n\) is the term number.
This formula makes it easy to find specific terms without listing all terms in the sequence.
Exploring Series
A series is simply the sum of the elements of a sequence. In the context of arithmetic sequences, we often want to find the sum of the first few terms, which is what is known as an arithmetic series.
To calculate the sum of an arithmetic series, you can use the following formula:
  • \(S_n = \frac{n}{2} (a_1 + a_n)\)
Where:
  • \(S_n\) is the sum of the first \(n\) terms,
  • \(a_1\) is the first term,
  • \(a_n\) is the \(n\)-th term, and
  • \(n\) is the number of terms to sum.
Understanding how to sum a series is critical in many mathematical applications, requiring careful alignment of terms and calculations to ensure accuracy.
Simplifying Mathematical Calculations
Mathematical calculations can often seem daunting, but breaking them down into manageable steps simplifies the process. Consider the exercise where we find the sum \(\sum_{k=1}^{4}(6k)\).
Here鈥檚 how you can simplify:
  • Identify what needs to be calculated. In this case, you were asked to sum four terms of the expression \(6k\).
  • Follow a step-by-step procedure. Substitute each \(k\) value from 1 to 4 and calculate the expression \(6k\) for each step.
  • Finally, sum up all individual outcomes: \(6 + 12 + 18 + 24\).
This illustrates the simplicity and power of structured calculations. Being systematic helps not only in arithmetic sequences and series but also in broader mathematics.

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Most popular questions from this chapter

Find the first six terms of the arithmetic sequence if the common difference is -3 and the ninth term is 10 .

Find the common difference of the arithmetic sequence with a first term of \(-164\) if its 36th term is \(-24.\)

Find each sum. $$ \sum_{k=3}^{5}\left(3 k^{2}-7\right) $$

Saving Money. Yasmeen puts \(\$ 60\) in a bank safety deposit box. She decides to begin a savings plan by putting \(\$ 50\) more in the box every month. Write the first six terms of an arithmetic sequence that gives the monthly amounts in her safety deposit box. Then find the amount of money that she will have in the box after 10 years of the deposits.

Write the first five terms of the geometric sequence if its first term is \(-64, r>0,\) and its fifth term is \(-4\).
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