Certainly! Let's go through the question step-by-step to find the value of the summation:
[tex]\[ \sum_{p=0}^5 (Y - X_i^p) \][/tex]
This summation notation represents the sum of the expression [tex]\( Y - X_i^p \)[/tex] as [tex]\( p \)[/tex] varies from [tex]\( 0 \)[/tex] to [tex]\( 5 \)[/tex].
### Step 1: Write Out Each Term in the Summation
First, we need to list each term explicitly from [tex]\( p = 0 \)[/tex] to [tex]\( p = 5 \)[/tex]:
[tex]\[
(Y - X_i^0), (Y - X_i^1), (Y - X_i^2), (Y - X_i^3), (Y - X_i^4), (Y - X_i^5)
\][/tex]
### Step 2: Sum All Terms
Now, let's sum all these individual terms together:
[tex]\[
(Y - X_i^0) + (Y - X_i^1) + (Y - X_i^2) + (Y - X_i^3) + (Y - X_i^4) + (Y - X_i^5)
\][/tex]
### Step 3: Simplify the Expression
Next, we'll simplify the expression by grouping the [tex]\( Y \)[/tex] terms together and the [tex]\( - X_i^p \)[/tex] terms together:
[tex]\[
Y + Y + Y + Y + Y + Y - (X_i^0 + X_i^1 + X_i^2 + X_i^3 + X_i^4 + X_i^5)
\][/tex]
Combine the [tex]\( Y \)[/tex] terms:
[tex]\[
6Y - (X_i^0 + X_i^1 + X_i^2 + X_i^3 + X_i^4 + X_i^5)
\][/tex]
### Step 4: Substitute the Powers of [tex]\( X_i \)[/tex]
We know that [tex]\( X_i^0 = 1 \)[/tex]. Substituting 1 for [tex]\( X_i^0 \)[/tex], we get:
[tex]\[
6Y - (1 + X_i + X_i^2 + X_i^3 + X_i^4 + X_i^5)
\][/tex]
Distribute the minus sign:
[tex]\[
6Y - 1 - X_i - X_i^2 - X_i^3 - X_i^4 - X_i^5
\][/tex]
### Step 5: Arrive at the Final Result
So, the summation of the given series simplifies to:
[tex]\[
\boxed{-X_i^5 - X_i^4 - X_i^3 - X_i^2 - X_i + 6Y - 1}
\][/tex]
This is the step-by-step solution to the given problem.
