Question

Which expression defines the given series for seven terms?

15 + 19 + 23 + . . .

Answer 1

The expression defines the given series for seven terms, i don't understand the question but i do know the sum which you should know to

15 + 19 + 23 = 57

Sorry

Answer 2

The expression is
`\sum _(n=1)^7\:\left(11+4n\right)` and sum of first seven terms is 189.

Explanation:

Given : The series 15 + 19 + 23 + . . .

We have to find an expression that defines the series for seven terms .

15 can be written as 11 + 4(1)

19 can be written as 11 + 4(2)

23 can be written as 11 + 4(3)

and so on

Thus, in general for 7 terms,

We can write it as
`\sum _(n=1)^7\:\left(11+4n\right)`

Thus,
`\sum _(n=1)^7\:\left(11+4n\right)`

Apply the sum rule,
`\sum a_n+b_n=\sum a_n+\sum b_n`

`=\sum _(n=1)^711+\sum _(n=1)^74n`

Consider,
`\sum _(n=1)^711`,

`\mathrm{Apply\:the\:Sum\:Formula:\quad }\sum _(k=1)^n\:a\:=\:a\cdot n`

`\sum _(n=1)^711=77`

Now, Consider
`\sum _(n=1)^74n`

`\mathrm{Apply\:the\:constant\:multiplication\:rule}:\quad \sum c\cdot a_n=c\cdot \sum a_n`

`=4\cdot \sum \:_(n=1)^7n`

`\sum _(n=1)^74n=112`

Thus,
`\sum _(n=1)^711+4n=189`

Thus, the expression is
`\sum _(n=1)^7\:\left(11+4n\right)` and sum of first seven terms is 189.