Answer:
Explanation:
The given expression is:
7 - 7x + 7x^2 - 7x^3 + ... + (-1)^n7x^n
To write this sum in sigma notation, we need to determine the pattern and the range of the summation.
Let's examine the terms in the expression:
Term 1: 7
Term 2: -7x
Term 3: 7x^2
Term 4: -7x^3
From this pattern, we can observe that the coefficient of each term alternates between 7 and -7. Additionally, the exponent of x increases by 1 for each subsequent term.
The general form of each term can be written as (-1)^(n-1) * 7 * x^(n-1), where n represents the term number.
Now, let's determine the range of the summation. The expression includes terms up to (-1)^n7x^n, which suggests that the sum should include terms from n = 1 to some value.
To represent the summation in sigma notation, we can use the following expression:
∑[n=1 to ∞] (-1)^(n-1) * 7 * x^(n-1)
Here, ∑ represents the summation symbol, n=1 represents the starting value of n, ∞ represents the ending value of n (which indicates an infinite sum), and the expression (-1)^(n-1) * 7 * x^(n-1) represents the general term.
In sigma notation, the given sum can be written as:
∑[n=1 to ∞] (-1)^(n-1) * 7 * x^(n-1)
This notation represents the sum of the given expression, where each term is obtained by substituting the appropriate value of n into the general term expression.