Certainly, let's solve the summation problem:
Problem:
Solve the summation:
ā_(n=1)^5 (C_(5,n) * i^n)
Solution:
Step 1: Understanding the summation:
* The summation symbol (ā) indicates that we need to add up a series of terms.
* The index variable is n, which starts at 1 and goes up to 5.
* Inside the summation, we have two parts:
* C_(5,n): This represents the binomial coefficient (also known as "5 choose n"). It is the number of ways to choose n items from a set of 5 items.
* i^n: This is the complex number i raised to the power of n.
Step 2: Calculating the terms:
Let's calculate each term of the summation for n from 1 to 5:
* n = 1: C_(5,1) * i^1 = 5i
* n = 2: C_(5,2) * i^2 = 10i^2
* n = 3: C_(5,3) * i^3 = 10i^3
* n = 4: C_(5,4) * i^4 = 5i^4
* n = 5: C_(5,5) * i^5 = i^5
Step 3: Adding the terms:
Now we add up all the terms:
5i + 10i^2 + 10i^3 + 5i^4 + i^5
Step 4: Simplifying the expression:
* Recall that i^2 = -1, i^3 = -i, and i^4 = 1.
* Substitute these values