Sure, let's break down the expression [tex]\( P = \sum_{k=1}^6 \left( k^2 - 3k + 2 \right) \)[/tex] step by step.
First, recognize that this is a summation:
[tex]\[ P = \sum_{k=1}^6 \left( k^2 - 3k + 2 \right) \][/tex]
This means we need to evaluate the expression [tex]\( k^2 - 3k + 2 \)[/tex] for each integer value of [tex]\( k \)[/tex] from 1 to 6 and then sum the results.
Let's compute the value of [tex]\( k^2 - 3k + 2 \)[/tex] for each [tex]\( k \)[/tex]:
1. For [tex]\( k = 1 \)[/tex]:
[tex]\[ 1^2 - 3 \cdot 1 + 2 = 1 - 3 + 2 = 0 \][/tex]
2. For [tex]\( k = 2 \)[/tex]:
[tex]\[ 2^2 - 3 \cdot 2 + 2 = 4 - 6 + 2 = 0 \][/tex]
3. For [tex]\( k = 3 \)[/tex]:
[tex]\[ 3^2 - 3 \cdot 3 + 2 = 9 - 9 + 2 = 2 \][/tex]
4. For [tex]\( k = 4 \)[/tex]:
[tex]\[ 4^2 - 3 \cdot 4 + 2 = 16 - 12 + 2 = 6 \][/tex]
5. For [tex]\( k = 5 \)[/tex]:
[tex]\[ 5^2 - 3 \cdot 5 + 2 = 25 - 15 + 2 = 12 \][/tex]
6. For [tex]\( k = 6 \)[/tex]:
[tex]\[ 6^2 - 3 \cdot 6 + 2 = 36 - 18 + 2 = 20 \][/tex]
Now, we sum these results:
[tex]\[ P = 0 + 0 + 2 + 6 + 12 + 20 \][/tex]
Summing these up:
[tex]\[ P = 0 + 0 + 2 = 2 \][/tex]
[tex]\[ P = 2 + 6 = 8 \][/tex]
[tex]\[ P = 8 + 12 = 20 \][/tex]
[tex]\[ P = 20 + 20 = 40 \][/tex]
Therefore, the value of [tex]\( P \)[/tex] is [tex]\( 40 \)[/tex].
