Answer:
The expression Sn - Sₙ₋₁ in an arithmetic sequence can be simplified to [1/2] * (a₁ + aₙ) - ((n-1)/2) * aₙ₋₁.
Explanation:
In an arithmetic sequence, Sn represents the sum of n terms. To find the expression Sn - Sₙ₋₁, we need to understand the formula for Sn and Sₙ₋₁.
The formula for Sn in an arithmetic sequence is given by:
Sn = (n/2) * (a₁ + aₙ),
where n represents the number of terms in the sequence, a₁ represents the first term, and aₙ represents the nth term.
Similarly, the formula for Sₙ₋₁ is:
Sₙ₋₁ = ((n-1)/2) * (a₁ + aₙ₋₁).
Now, let's substitute these formulas into the expression Sn - Sₙ₋₁:
Sn - Sₙ₋₁ = [(n/2) * (a₁ + aₙ)] - [((n-1)/2) * (a₁ + aₙ₋₁)].
Expanding and simplifying this expression, we get:
Sn - Sₙ₋₁ = (n/2) * (a₁ + aₙ) - ((n-1)/2) * (a₁ + aₙ₋₁).
Further simplifying, we have:
Sn - Sₙ₋₁ = (n/2) * a₁ + (n/2) * aₙ - ((n-1)/2) * a₁ - ((n-1)/2) * aₙ₋₁.
Combining like terms, we get:
Sn - Sₙ₋₁ = [(n/2) - ((n-1)/2)] * a₁ + [(n/2) - ((n-1)/2)] * aₙ - ((n-1)/2) * aₙ₋₁.
Simplifying further, we have:
Sn - Sₙ₋₁ = [1/2] * a₁ + [1/2] * aₙ - ((n-1)/2) * aₙ₋₁.
So, the expression Sn - Sₙ₋₁ can be simplified to:
Sn - Sₙ₋₁ = [1/2] * (a₁ + aₙ) - ((n-1)/2) * aₙ₋₁.
To summarize:
- The expression Sn - Sₙ₋₁ in an arithmetic sequence can be simplified to [1/2] * (a₁ + aₙ) - ((n-1)/2) * aₙ₋₁.