Question

find the sum
\\\sum_{n=51}^{90} 3n =\\

Explanation:

Step1: Identify the series type

The given expression is an arithmetic series:

$$\sum_{n=51}^{90} 3n$$

Step2: Determine the number of terms

Find the number of terms \(k\):

$$k = 90 - 51 + 1 = 40$$

Step3: Find the first and last terms

Calculate the first term \(a_1\) and last term \(a_{40}\):

$$a_1 = 3(51) = 153$$

$$a_{40} = 3(90) = 270$$

Step4: Calculate the sum

Use the arithmetic series sum formula:

$$S_{40} = \frac{40}{2} (153 + 270) = 20 \times 423 = 8460$$

Answer:

8460