To determine the correct summation formula that represents the series [tex]\( 13 + 9 + 5 + 1 \)[/tex], let's break down the analysis step-by-step.
### Step 1: Identify the Pattern and Key Parameters
First, let's identify the pattern of the arithmetic sequence:
- The series given is [tex]\( 13, 9, 5, 1 \)[/tex].
- The first term ([tex]\( a \)[/tex]) is [tex]\( 13 \)[/tex].
- The common difference ([tex]\( d \)[/tex]) is calculated as the difference between consecutive terms. Here, [tex]\( 9 - 13 = -4 \)[/tex]. Hence, the common difference [tex]\( d = -4 \)[/tex].
### Step 2: Arithmetic Sequence Formula
The nth term ([tex]\( T_n \)[/tex]) of an arithmetic sequence is given by:
[tex]\[ T_n = a + (n-1) \cdot d \][/tex]
For our sequence:
[tex]\[ T_n = 13 + (n-1) \cdot (-4) = 13 - 4(n-1) \][/tex]
Simplifying, we get:
[tex]\[ T_n = 13 - 4n + 4 \][/tex]
[tex]\[ T_n = -4n + 17 \][/tex]
### Step 3: Formulating the Summation Notation
Now we write the summation notation using the formula for the nth term we derived. We know there are 4 terms in the series:
So, the summation notation for this series from [tex]\( n = 1 \)[/tex] to [tex]\( n = 4 \)[/tex] will be:
[tex]\[
\sum_{n=1}^4 (-4n + 17)
\][/tex]
### Conclusion
Hence, the summation formula that represents the given series [tex]\( 13 + 9 + 5 + 1 \)[/tex] is:
[tex]\[
\sum_{n=1}^4 (-4n + 17)
\][/tex]
Other given options:
[tex]\[
\sum_{n=1}^4 (-4n - 15)
\][/tex]
and
[tex]\[
\sum_{n=13}^{16}(n-4)
\][/tex]
do not correctly represent this series.
Therefore, the correct answer is:
[tex]\[
\sum_{n=1}^4 (-4n + 17)
\][/tex]
