To solve the summation [tex]\(\sum_{j=10}^{15}(2j - 7)\)[/tex], we will evaluate the expression [tex]\(2j - 7\)[/tex] for each integer value of [tex]\(j\)[/tex] from 10 to 15 and then sum the results.
1. Substitute [tex]\( j = 10 \)[/tex] into [tex]\( 2j - 7 \)[/tex]:
[tex]\[
2(10) - 7 = 20 - 7 = 13
\][/tex]
2. Substitute [tex]\( j = 11 \)[/tex] into [tex]\( 2j - 7 \)[/tex]:
[tex]\[
2(11) - 7 = 22 - 7 = 15
\][/tex]
3. Substitute [tex]\( j = 12 \)[/tex] into [tex]\( 2j - 7 \)[/tex]:
[tex]\[
2(12) - 7 = 24 - 7 = 17
\][/tex]
4. Substitute [tex]\( j = 13 \)[/tex] into [tex]\( 2j - 7 \)[/tex]:
[tex]\[
2(13) - 7 = 26 - 7 = 19
\][/tex]
5. Substitute [tex]\( j = 14 \)[/tex] into [tex]\( 2j - 7 \)[/tex]:
[tex]\[
2(14) - 7 = 28 - 7 = 21
\][/tex]
6. Substitute [tex]\( j = 15 \)[/tex] into [tex]\( 2j - 7 \)[/tex]:
[tex]\[
2(15) - 7 = 30 - 7 = 23
\][/tex]
Next, we sum all these results:
[tex]\[
13 + 15 + 17 + 19 + 21 + 23
\][/tex]
Adding these terms sequentially, we get:
[tex]\[
13 + 15 = 28
\][/tex]
[tex]\[
28 + 17 = 45
\][/tex]
[tex]\[
45 + 19 = 64
\][/tex]
[tex]\[
64 + 21 = 85
\][/tex]
[tex]\[
85 + 23 = 108
\][/tex]
Thus, the value of the summation is:
[tex]\[
\sum_{j=10}^{15} (2j - 7) = 108
\][/tex]
