Question

Which expression defines the given series for seven terms? 15 + 19 + 23 + . . .

Answer 1

Answer:

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

Step-by-step 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.

Answer 2

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