Question

06:posttest hs: algebra 2b m (sequential) / 6:sequences and series 30. which expression defines the given series for seven terms? 15 + 19 + 23 +... options: \\(\sum_{n=1}^{7} (11 + 4n)\\), \\(\sum_{n=1}^{7} (13 + 2n)\\), \\(\sum_{n=1}^{7} (9 + 6n)\\), \\(\sum_{n=1}^{7} (10 + 5n)\\)

Explanation:

Step1: Identify series type and parameters

The series $15+19+23+\dots$ is arithmetic, with first term $a_1=15$, common difference $d=19-15=4$.

Step2: Find nth term formula

The nth term of an arithmetic sequence is $a_n=a_1+(n-1)d$. Substitute values:
$a_n=15+(n-1)4=15+4n-4=11+4n$

Step3: Write summation for 7 terms

The summation for 7 terms is $\sum_{n=1}^{7}(11 + 4n)$

Step4: Verify with first term

Substitute $n=1$: $11+4(1)=15$, matches the first term.
Substitute $n=2$: $11+4(2)=19$, matches the second term.

Answer:

$\sum_{n=1}^{7}(11 + 4n)$