Question

Find the sum of the summation of 3 i minus 15, from i equals 2 to 7.

Answer 1

`\sum _(i=2)^73i-15=-9`

Explanation:

Given :
`\sum _(i=2)^73i-15`

We have to evaluate the given summation.

Consider
`\sum _(i=2)^73i-15`

using, we have,

`\sum _(k=m)^n\:=\:\sum _(k=1)^n\:-\:\sum _(k=1)^(m-1)`

`=\sum _(i=1)^73i-15-\sum _(i=1)^13i-15` ..............(1)

Consider,
`\sum _(i=1)^73i-15`

Apply sum rule,

`\quad \sum a_n+b_n=\sum a_n+\sum b_n` , we have,

`=\sum _(i=1)^73i-\sum _(i=1)^715`

Now, first consider,
`\sum _(i=1)^73i`

Apply constant multiplication rule,
`\quad \sum c\cdot a_n=c\cdot \sum a_n` we have,

`=3\cdot \sum \:_(i=1)^7i`

`=3\cdot \:28\\\\=84`

Also,
`\sum _(i=1)^715=105`

Thus,
`\sum _(i=1)^73i-15=84-105=-21`

Similarly,
`\sum _(i=1)^13i-15=-12`

Substitute in (1) , we have,

Thus,
`\sum _(i=2)^73i-15=-21-(-12)=-21+12=-9`

Thus,
`\sum _(i=2)^73i-15=-9`

Answer 2

The summation symbol means we use the variable i in the equation from the given range. In this case, we are given with the equation 3 i - 15 and i ranges from 2 to 7. The summation using simply the calculator is equal to -9. The answer to this problem is -9.