Answer :

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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]

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