Question

For each of the four series (a)-(d), identify which have the same number of terms and which have the same value. (a) \(\sum_{i=4}^{20} i\) (b) \(\sum_{j=1}^{16}(-j)\) (c) \(\quad 0-1-2-\ldots-16\) (d) \(\quad 48+50+52+54\)

Short answer

Answer: Series (a) and (c) have the same number of terms (17 terms), and series (b) and (d) have the same value (204).

Step-by-step solution

Find the number of terms of each series.

For series (a), we can find the number of terms by taking the difference between the upper and lower limits and adding 1: Number of terms in (a) = 20 - 4 + 1 = 17. For series (b), we follow the same procedure: Number of terms in (b) = 16 - 1 + 1 = 16. For series (c), we can count the terms directly, there are 17 terms from 0 to -16. For series (d), we can find the number of terms by the difference between the last and first term, divided by the common difference (which is 2), and then adding 1: Number of terms in (d) = (54 - 48) / 2 + 1 = 4.

Calculate the value of each series.

For series (a), we will use the arithmetic series sum formula \(\frac{n(a_1 + a_n)}{2}\), where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term: Value of (a) = (17(4 + 20)) / 2 = 204. For series (b), we will use the same formula but with a negative common difference: Value of (b) = (16(-1 - 16)) / 2 = -136. For series (c), we can simply add the terms: Value of (c) = -1 - 2 - ... - 16 = -136. For series (d), we can add the terms: Value of (d) = 48 + 50 + 52 + 54 = 204.

Compare the properties of each series.

After calculating the number of terms and value for each series, we can now compare them: Series (a) and (c) have the same number of terms (17 terms). Series (b) and (d) have the same value (204). Therefore, series (a) and (c) have the same number of terms, and series (b) and (d) have the same value.

Key concepts

Number of Terms in an Arithmetic Series

In an arithmetic series, the number of terms plays a central role in understanding its structure. To determine the number of terms, you need to consider the starting and ending terms of the series. For instance, in series (a), it begins at 4 and ends at 20. Here, you calculate the number of terms by taking the difference between the upper and lower limits and adding 1:
- Number of terms in (a) = 20 - 4 + 1 = 17.
This method is applied similarly to series (b), resulting in 16 terms because it starts at 1 and ends at 16:
- Number of terms in (b) = 16 - 1 + 1 = 16.
Series (c) explicitly lists all terms from 0 to -16, which you can simply count:
- Number of terms in (c) = 17 terms.
Finally, series (d) involves using the common difference to find the number of terms, starting at 48 and ending at 54. Here, the common difference is 2:
- Number of terms in (d) = (54 - 48) / 2 + 1 = 4.

Series Value and Sum Formula

To find the total value of an arithmetic series, you can apply the arithmetic series sum formula. This formula is shown as \(S_n = \frac{n(a_1 + a_n)}{2}\), where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.
For series (a), the sum becomes:
- Value of (a) = \(\frac{17(4 + 20)}{2} = 204\).
In contrast, for series (b) which has a negative common difference, you find the sum using similar steps:
- Value of (b) = \(\frac{16(-1 - 16)}{2} = -136\).
With series (c), since it's already laid out, adding each term gives:
- Value of (c) = -1 - 2 - ... - 16 = -136.
Lastly, series (d) can be summed directly:
- Value of (d) = 48 + 50 + 52 + 54 = 204.

Understanding Common Difference

The common difference is crucial in arithmetic series as it reveals the constant interval between consecutive terms. For example, in series (a) and series (b), the common difference is 1 and -1, respectively. This indicates each term increases by 1 in series (a) and decreases by 1 in series (b).
- Series (a): First term 4, last term 20, common difference = 1. - Series (b): Common difference = -1.
Series (c) decreases by 1 with each term starting from 0 to -16. Similarly, series (d) showcases a positive common difference of 2:
- Series (d): 48, 50, 52, 54, common difference = 2.
This common difference provides a pattern for the sequence, enabling easier calculation of terms and series values.