Question

Find the \(30^{\text {th }}\) positive multiple of 6 and the sum of the first 30 positive multiples of 6 .

Short answer

The 30th positive multiple of 6 is 180, and the sum of the first 30 positive multiples of 6 is 2790.

Step-by-step solution

Find the Sequence of Positive Multiples of 6

The sequence can be listed by multiplying the numbers 1 through 30 by 6. This can be constructed using a general formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\). In our case, \(a_1=6\) and \(d=6\). Therefore, the formula for nth term of the sequence is \(a_n = 6 + 6(n-1)\).

Find the 30th Positive Multiple of 6

Using the formula found in Step 1, we will plug in n = 30 to find the 30th positive multiple of 6: \(a_{30} = 6 + 6(30 - 1)\). This results in \(a_{30} = 6 + 6(29) = 6 + 174 = 180\). So, the 30th positive multiple of 6 is 180.

Find the Sum of the First 30 Positive Multiples of 6

To find the sum of the sequence, we will use the arithmetic series sum formula: sum = (number of terms / 2) * (first term + last term). In this case, we have the first term (\(a_1=6\)), the last term (\(a_{30}=180\)), and the number of terms (n = 30). Plugging these into the formula, we get the sum: Sum = \(\frac{30}{2}(6+180) = 15(186) = 2790\) The sum of the first 30 positive multiples of 6 is 2790.

Key concepts

Multiples of 6

A multiple of 6 is a result you get when you multiply 6 by any positive integer. For example:

• 6 × 1 = 6
• 6 × 2 = 12
• 6 × 3 = 18

Each product in this list is a multiple of 6.
When you are asked to find a specific multiple, such as the 30th multiple of 6, you simply multiply 6 by 30.
In general, if you want the nth multiple of 6, you would compute it using the formula: 6n. This way, finding multiples becomes a straightforward task!

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence.
An arithmetic sequence is a sequence of numbers with a common difference between consecutive terms. This means you add the same number to go from one term to the next.
For example, in the sequence of multiples of 6 like 6, 12, 18, ..., the difference is 6.

• First term, \(a_1 = 6\)
• Common difference, \(d = 6\)

To form a series, you add up the terms in the sequence. If you list the first 30 multiples of 6, you can add them to form an arithmetic series.

Sum of Sequence

Calculating the sum of a sequence means finding the total when you add all the numbers in a sequence.
For arithmetic sequences, there's a handy formula: sum = \(\frac{n}{2}(a_1 + a_n)\). This helps to quickly find the sum without adding each number by hand.

• \(n\) is the number of terms.
• \(a_1\) is the first term.
• \(a_n\) is the last term.

So for the first 30 multiples of 6, you take 30 terms, the first term as 6, and the last term as 180 (since that's the 30th term), and plug them into the formula.
This results in the sum \(2790\), an efficient way to calculate the total of many numbers in a systematic way.