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

  1. $0$
  2. $-1$
  3. $1$
  4. Not unique
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Sum of given  n  terms in AP  is =   $n/2(2a+(n-1)d)$ , where n = no of term , a = intial term value , d = common difference

according to given question   sum of 11 terms in AP =  sum of 19 terms in AP

                                         $11/2(2a+10d) = 19/2(2a+18d)$

                                         $22a+110d=38a+342d$

                                        $16a+232d =0$ ------ (1)

similar  sum of 30 term in AP = $30/2(2a+29d)$

    multiply by 8 this , we get  =   $30/2(16a+232d)$

    from equation 1 ,  sum of 30 term in AP = $30/2(0)$ = $0$

So ans is option 1) =0
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