Question

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ 2,4,6,8, \ldots $$

Short answer

The sequence is arithmetic with a common difference of 2. The sum of the first 50 terms is 2550.

Step-by-step solution

Identify the sequence type

Compare the differences between consecutive terms: \(4-2 = 2\), \(6-4 = 2\), \(8-6 = 2\). Since the differences are constant, the sequence is arithmetic.

Find the common difference

The common difference (\(d\)) is the difference between consecutive terms. Here, \(d = 4 - 2 = 2\).

Calculate the 50th term

Use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1)d\) where \(a_1 = 2\), \(d = 2\), and \(n = 50\). Thus,\(a_{50} = 2 + (50-1) \times 2 = 2 + 98 = 100\).

Find the sum of the first 50 terms

Use the sum formula for the first \(n\) terms of an arithmetic sequence: \(S_n = \frac{n}{2}(a_1 + a_n)\). Substitute \(n = 50\), \(a_1 = 2\), and \(a_n = 100\): \(S_{50} = \frac{50}{2}(2 + 100) = 25 \times 102 = 2550\).

Key concepts

Common Difference

The common difference is what defines an arithmetic sequence. It's the constant amount you add (or subtract) to get from one term to the next. In the given sequence, let's check whether we have a common difference:
2, 4, 6, 8, ...
The difference between each pair of terms is:

• 4 - 2 = 2
• 6 - 4 = 2
• 8 - 6 = 2

Since the difference is always 2, we can conclude this sequence is arithmetic.
The common difference is a fundamental property of an arithmetic sequence. It helps determine many other aspects, like the next term in the sequence and the sum of terms.

Sum of Terms

Summing the terms of an arithmetic sequence is straightforward once you know the formula. The sum of the first n terms (denoted as S_n) can be calculated using:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Let's break it down using our example sequence:
The first term \( a_1 = 2 \). The nth term here is the 50th term, which is 100 (found by plugging n = 50 into the nth term formula).So, n = 50, a_1 = 2, and a_n = 100.
Now substitute these into the formula:\[ S_{50} = \frac{50}{2}(2 + 100) = 25 \times 102 = 2550 \]Thus, the sum of the first 50 terms is 2550.
Understanding the sum formula is crucial because it simplifies complex additions and helps in quickly determining the sum of large sequences.

Sequence Identification

Identifying whether a sequence is arithmetic or geometric (or neither) is the starting point. You can do this by checking the difference or ratio between consecutive terms:
For an arithmetic sequence, the difference between terms is constant. For our example:

• 4 - 2 = 2
• 6 - 4 = 2
• 8 - 6 = 2

Since the difference is constant, we conclude it is an arithmetic sequence.
For a geometric sequence, you’d check if the ratio between terms is constant. This is done by dividing subsequent terms.In general, identifying the type of sequence helps in knowing which formulas to use for finding terms, sums, and other properties.