To find the sum given by [tex]\( \sum_{j=1}^{10} (3j + 4) \)[/tex], we need to add up the values of the expression [tex]\( 3j + 4 \)[/tex] for [tex]\( j \)[/tex] ranging from 1 to 10. Here's a step-by-step breakdown of how we can do this:
1. Write out the series:
We need to calculate the sum [tex]\( (3 \cdot 1 + 4) + (3 \cdot 2 + 4) + (3 \cdot 3 + 4) + \cdots + (3 \cdot 10 + 4) \)[/tex].
2. Compute each term individually:
[tex]\[
\begin{aligned}
& 3 \cdot 1 + 4 = 3 + 4 = 7, \\
& 3 \cdot 2 + 4 = 6 + 4 = 10, \\
& 3 \cdot 3 + 4 = 9 + 4 = 13, \\
& 3 \cdot 4 + 4 = 12 + 4 = 16, \\
& 3 \cdot 5 + 4 = 15 + 4 = 19, \\
& 3 \cdot 6 + 4 = 18 + 4 = 22, \\
& 3 \cdot 7 + 4 = 21 + 4 = 25, \\
& 3 \cdot 8 + 4 = 24 + 4 = 28, \\
& 3 \cdot 9 + 4 = 27 + 4 = 31, \\
& 3 \cdot 10 + 4 = 30 + 4 = 34.
\end{aligned}
\][/tex]
3. Form the sum of these computed values:
[tex]\[
7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 + 34.
\][/tex]
4. Add the values together:
[tex]\[
7 + 10 = 17, \\
17 + 13 = 30, \\
30 + 16 = 46, \\
46 + 19 = 65, \\
65 + 22 = 87, \\
87 + 25 = 112, \\
112 + 28 = 140, \\
140 + 31 = 171, \\
171 + 34 = 205.
\][/tex]
Therefore, the sum [tex]\( \sum_{j=1}^{10} (3j + 4) \)[/tex] is [tex]\( 205 \)[/tex].
