Question

Find the sum of the given arithmetic series.

27 + 54 + 81 + 108 + ... + 540

Answer 1

To find the sum of the arithmetic series \(27 + 54 + 81 + 108 + ... + 540\), we first need to identify the first term, common difference, and the number of terms in the series.

1. **Identifying the first term**:
The first term \(a = 27\).

2. **Finding the common difference**:
The common difference \(d\) can be calculated as follows:
\[
d = 54 - 27 = 27.
\]

3. **Identifying the last term**:
The last term of the series is given as \(l = 540\).

4. **Finding the number of terms**:
We can use the formula for the \(n\)-th term of an arithmetic series, given by:
\[
a_n = a + (n-1)d.
\]
We know \(a_n = 540\), \(a = 27\), and \(d = 27\). Plugging these values into the formula gives:
\[
540 = 27 + (n-1) \cdot 27.
\]
We can simplify this equation:
\[
540 - 27 = (n-1) \cdot 27,
\]
\[
513 = (n-1) \cdot 27.
\]
Dividing both sides by 27:
\[
n - 1 = \frac{513}{27} = 19.
\]
Therefore:
\[
n = 19 + 1 = 20.
\]

5. **Calculating the sum of the arithmetic series**:
The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic series is:
\[
S_n = \frac{n}{2} (a + l).
\]
Now substituting \(n = 20\), \(a = 27\), and \(l = 540\):
\[
S_{20} = \frac{20}{2} \cdot (27 + 540).
\]
Simplifying this:
\[
S_{20} = 10 \cdot (27 + 540) = 10 \cdot 567 = 5670.
\]

Thus, the sum of the given arithmetic series is \(\boxed{5670}\).