If in an AP (S_{18}=810) and (S_9=270), what is the sum from the (10)th term to the (18)th term?

(540)

The sum from the 10th term through the 18th term can be obtained by removing the first nine terms from the sum of the first eighteen terms. In symbols, the required range is \(a_{10}+a_{11}+\cdots+a_{18}=S_{18}-S_9\). The given values are \(S_{18}=810\) and \(S_9=270\), so substitution gives \(810-270=540\). There are nine terms in this range, but no common difference or individual terms are needed because the two partial sums already contain exactly the required information.

Thus option D, 540, is correct. A frequent mistake is to subtract \(S_8\), which would include only terms 9 through 18, or to subtract \(S_{10}\), which would remove the 10th term as well. Since the requested sequence begins immediately after the first nine terms, \(S_{18}-S_9\) is the precise calculation.

This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the sum of the first $n$ terms of an AP.