If the sum of first 11
terms of an arithmetic progression equals that of first 19 terms, then what is
the sum of first 30 terms?

(1) 0

(2) -1

(3) 1

(4) None of these

For Answer
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Sum of first n terms of a
n A.P = S

_{n}= n/2 (2a+(n-1)d), where a is first term and d is common difference
S

_{11}= S_{19}=> 11/2(2a+10d) = 19/2(2a+18d)
=> 11(a+5d) = 19(a+9d)

=> 11a+55d = 19a+171d

=> 8a+116d = 0

=> 2a+29d = 0

Sum of first 30 terms of A.P = S

_{30}= 30/2 (2a+29d) = 0**Answer (1)**

We can generalize this as:

For an arithmetic progression, If sum of first ‘m’ terms = sum of first ‘n’
terms, then sum of first ‘m+n’ terms = 0
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