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In an arithmetic sequence of terms , Sn represents sum of n terms , then what is Sn - S ₙ₋₁

User Wwarby
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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.

User Tu Nguyen
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