Sum Evaluation of 9 Series

Evaluating the Sum of a Series

Given Problem Statement

Evaluate the sum represented by the following expression:
extEvaluate:9∑k=19(5k+8).ext{Evaluate: } 9 \sum_{k=1}^{9} (5k + 8).

Components of the Sum

The expression $(5k + 8)$ is a linear function of $k$. In the summation, we need to calculate this expression for each integer value of $k$ from 1 to 9 and then sum all these results together.

Step-by-Step Calculation

To evaluate the sum, we will break it down into two parts: the part that involves $5k$ and the constant part, $8$.

Step 1: Break the Summation

We can rewrite the sum as the sum of two separate series:
∑k=19(5k+8)=∑k=195k+∑k=198.\sum_{k=1}^{9} (5k + 8) = \sum_{k=1}^{9} 5k + \sum_{k=1}^{9} 8.

Step 2: Calculate Each Part

  1. Calculate ∑k=195k\sum_{k=1}^{9} 5k
    Using the property of summation, we can factor out the constant:
    ∑k=195k=5∑k=19k.\sum_{k=1}^{9} 5k = 5 \sum_{k=1}^{9} k.
    The formula for the sum of the first $n$ natural numbers is given by:
    ∑k=1nk=n(n+1)2.\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}.
    For $n = 9$:
    ∑k=19k=9(9+1)2=9×102=45.\sum_{k=1}^{9} k = \frac{9(9 + 1)}{2} = \frac{9 \times 10}{2} = 45.
    Therefore:
    ∑k=195k=5×45=225.\sum_{k=1}^{9} 5k = 5 \times 45 = 225.

  2. Calculate ∑k=198\sum_{k=1}^{9} 8
    Since 8 is a constant, this sum can be calculated as:
    ∑k=198=8×9=72.\sum_{k=1}^{9} 8 = 8 \times 9 = 72.

Step 3: Combine Results

Now we combine the results of the two parts:
∑k=19(5k+8)=225+72=297.\sum_{k=1}^{9} (5k + 8) = 225 + 72 = 297.

Final Result

Therefore, the value of the sum is:
Final result: 297.\text{Final result: } 297.