Question

determine the value of the sum below.

\\\sum_{k=4}^{9} (3k - 2)\\

Explanation:

Expand the terms of the summation

Using the Sigma Notation knowledge point

$$ \sum_{k=4}^{9} (3k - 2) $$

The index \(k\) ranges from \(4\) to \(9\). We evaluate the term \(3k - 2\) for each integer in this range:

• For \(k = 4\): \(3(4) - 2 = 10\)
• For \(k = 5\): \(3(5) - 2 = 13\)
• For \(k = 6\): \(3(6) - 2 = 16\)
• For \(k = 7\): \(3(7) - 2 = 19\)
• For \(k = 8\): \(3(8) - 2 = 22\)
• For \(k = 9\): \(3(9) - 2 = 25\)

Identify the series type

Using the Arithmetic Series knowledge point
The sequence of terms is:

$$ 10, 13, 16, 19, 22, 25 $$

This is an arithmetic sequence with:

• First term \(a_1 = 10\)
• Last term \(a_n = 25\)
• Number of terms \(n = 9 - 4 + 1 = 6\)

Calculate the sum of the series

Using the Summation Formula knowledge point

$$ S_n = \frac{n}{2}(a_1 + a_n) $$

$$ S_6 = \frac{6}{2}(10 + 25) = 3 \times 35 = 105 $$

Answer:

105