Problem 1 - Algebra 2

Question

Find the sum of each arithmetic series. $$ 5+11+17+\dots+95 $$

Step-by-Step Solution

Answer

The sum of the series is 800.

Step 1: Identify the first term (a)

The first term of the sequence is the initial number given in the series. Here, the first term \(a\) is 5.

Step 2: Determine the common difference (d)

To find the common difference \(d\), subtract the first term from the second term: \(d = 11 - 5 = 6\).

Step 3: Identify the last term (l)

The last term of the series is given as 95. Thus, \(l = 95\).

Step 4: Find the number of terms (n)

Use the formula for the last term of an arithmetic sequence: \(l = a + (n-1) \, d\).Substitute the values: \(95 = 5 + (n-1) \, 6\).Solve for \(n\) to find the number of terms in the series.\(95 = 5 + 6n - 6\),\(96 = 6n\),\(n = 16\).

Step 5: Calculate the sum of the arithmetic series

Use the sum formula for an arithmetic series: \(S_n = \frac{n}{2} (a + l)\).Substitute the known values: \(S_{16} = \frac{16}{2} (5 + 95)\).This simplifies to \(S_{16} = 8 \times 100\),So, \(S_{16} = 800\).

Key Concepts

Common Difference

In an arithmetic sequence, each term after the first is found by adding a constant value, known as the common difference, to the previous term. The common difference is denoted by \(d\). This difference is crucial as it determines how the sequence progresses. For example, consider the sequence: 5, 11, 17, ... In this series, the common difference \(d\) is found by subtracting the first term from the second term: \(d = 11 - 5 = 6\).

• It ensures that the sequence progresses evenly.
• Without the common difference, the pattern of the sequence wouldn't be consistent.

Recognizing the common difference helps in predicting any term of the sequence and calculating the sum of the series.

Sum of Series

The sum of an arithmetic series is the total of all terms added together. To find this sum, there is a handy formula: \(S_n = \frac{n}{2} (a + l)\) where \(S_n\) represents the sum of \(n\) terms, \(a\) is the first term, and \(l\) is the last term.

• This formula leverages the symmetry of the sequence.
• By effectively pairing terms from the start and end, it simplifies the sum calculation.

In the example series 5, 11, 17, ..., 95, knowing that there are 16 terms, the calculation is \(S_{16} = \frac{16}{2} (5 + 95)\). This calculation results in the sum \(S_{16} = 800\).
Using the sum formula is efficient and straightforward compared to adding each term manually.

Number of Terms Formula

To find the number of terms in an arithmetic sequence, you need a formula that links the last term, the first term, and the common difference. This is often referred to as the formula for the last term: \(l = a + (n-1) \cdot d\), where \(l\) is the last term, \(a\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.

• This formula helps you solve for \(n\), the total number of terms in the sequence.
• The equation rearranges to solve for \(n\) when \(l\), \(a\), and \(d\) are known.

Applying it to the series 5, 11, 17, ..., 95, gives: \(95 = 5 + (n-1) \cdot 6\), eventually simplifying to \(n = 16\).
This formula is invaluable for determining the length of the sequence when the pattern and boundaries are set.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two successive terms is a constant. This consistency allows us to predict subsequent terms and implement calculations for sums and term positions.

• An arithmetic sequence can be visualized or thought of as a set of numbers increasing (or decreasing) by the same amount each time.
• It provides a straightforward method to illustrate linear growth or decay.

For example, in the sequence 5, 11, 17, ..., the common conception is the addition of 6 to generate the next term.
Understanding arithmetic sequences is fundamental to grasping the wider topic of arithmetic series, where the goal often includes finding sums or identifying specific terms within the sequence.