Find the sum of the following arithmetic series:
3plus9plus15plus21plus...plus81.
The sum of this series is
The series is an arithmetic series with first term \(a = 3\), common difference \(d = 6\), and last term \(l = 81\).
The formula for the sum of an arithmetic series is:
\[ S = \frac{n}{2}(a + l) \]
where \( n \) is the number of terms in the series.
To find the number of terms in the series, we can use the formula for the nth term of an arithmetic series:
\[ l = a + (n-1)d \]
Substitute the known values:
\[ 81 = 3 + (n-1)6 \]
\[ 81 = 3 + 6n - 6 \]
\[ 78 = 6n \]
\[ n = 13 \]
Now, plug in the values for \( a \), \( l \), and \( n \) into the formula for the sum:
\[ S = \frac{13}{2}(3 + 81) \]
\[ S = \frac{13}{2}(84) \]
\[ S = 546 \]
Therefore, the sum of the arithmetic series 3 + 9 + 15 + 21 + ... + 81 is 546.