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