Question

For each of the four series (a)-(d), identify which have the same number of terms and which have the same value. (a) \(\sum_{i=1}^{20} 3\) (b) \(\sum_{j=1}^{20}(3 j)\) (c) \(0+3+6 \ldots+60\) (d) \(60+57+54+\cdots+6+3\)

Short answer

Answer: Series (a), (b), and (d) have the same number of terms (20 terms), while series (b) and (c) have the same value (630).

Step-by-step solution

Determine the number of terms in series (c) and (d)

To find the number of terms in series (c) and (d), we can use the arithmetic sequence formula: \(n = \frac{(L-A)}{D} + 1\) where \(n\) is the number of terms, \(A\) is the first term, \(D\) is the common difference, and \(L\) is the last term. For series (c), the first term (\(A\)) is 0, the common difference (\(D\)) is 3, and the last term (\(L\)) is 60. For series (d), the first term (\(A\)) is 60, the common difference (\(D\)) is -3, and the last term (\(L\)) is 3. Let's calculate the number of terms for both series now.

Calculate the number of terms for series (c)

For series (c): \(n = \frac{(60 - 0)}{3} + 1\) \(n = \frac{60}{3} + 1\) \(n = 20+1\) \(n = 21\) So, series (c) has 21 terms.

Calculate the number of terms for series (d)

For series (d): \(n = \frac{(3 - 60)}{-3} + 1\) \(n = \frac{-57}{-3} + 1\) \(n = 19+1\) \(n = 20\) So, series (d) has 20 terms.

Compare the number of terms

Now we have the number of terms for all series: (a) 20 terms (b) 20 terms (c) 21 terms (d) 20 terms Series (a), (b), and (d) have the same number of terms (20 terms).

Calculate the value of each series

Next, let's calculate the value of each series: (a) \(\sum_{i=1}^{20} 3 = 20 \cdot 3 = 60\) (b) \(\sum_{j=1}^{20}(3 j) = 3 \left(\sum_{j=1}^{20} j\right) = 3 \frac{20 (20+1)}{2} = 3 \cdot 210 = 630\) (c) \(0 + 3 + 6 + \cdots + 60 = \frac{21(0+60)}{2} = \frac{21 \cdot 60}{2} = 21 \cdot 30 = 630\) (d) \(60 + 57 + 54 + \cdots + 6 + 3 = 20 \cdot 3 - (0+3+6+ \cdots + 57) = 60 - 630 + 60 = -510\)

Compare the values of each series

Now we have the value of each series: (a) Value = 60 (b) Value = 630 (c) Value = 630 (d) Value = -510 Series (b) and (c) have the same value (630). In conclusion, series (a), (b), and (d) have the same number of terms, while series (b) and (c) have the same value.

Key concepts

Arithmetic Sequences

When we talk about arithmetic sequences, we refer to a sequence of numbers where each term after the first is the sum of the previous term plus a constant known as the common difference. This common difference can be positive, negative, or zero. Understanding an arithmetic sequence is crucial in solving many mathematical problems involving series and sequences.

To identify an arithmetic sequence, look for:

• A constant difference between consecutive terms.
• A formula for any term ( T_n = A + (n-1)D ), where A is the first term, D is the common difference, and n is the term number.

This pattern is what gives arithmetic sequences a uniform structure. Whether you start small or big, the increase or decrease in steps will always be the same throughout.

Sum of Series

The sum of a series is essentially the total when you add all the terms in a sequence. For arithmetic series, there’s a nifty formula to get this sum without having to add each number individually. This saves a lot of time, especially for long sequences.

The formula is: \( S_n = \frac{n}{2}(A + L) \) , where:

• S_n is the sum of the series.
• n is the number of terms.
• A is the first term.
• L is the last term.

Using this formula, you can find the sum of an arithmetic sequence efficiently. Remember, this applies when the sequence is clearly arithmetic, meaning it must have equal intervals between terms. This ensures that each pair of first and last terms averages out across the sequence.

Number of Terms in a Sequence

Determining the number of terms in an arithmetic sequence is a vital skill. It helps you understand how long your sequence is and prepares you for calculations involving the sequence.

The essential formula is: \( n = \frac{(L-A)}{D} + 1 \) , where:

• L is the last term.
• A is the first term.
• D is the common difference.

By rearranging the arithmetic sequence formula, you can derive this to help find the exact number of terms. This aspect can come in handy when working with sequences presented in different forms, such as a sum or partial list. Moreover, understanding this concept lays the groundwork for better managing and calculating series sums and terms.