Let An be a sequence defined by A1=1 then A1+ 2A2 + 3A3+ + (n-1)A n-1 - Maths - Sequences and Series

Question

Let An be a sequence defined by A1=1 then A1+ 2A2 + 3A3+ + (n-1)A
n-1 = (n^2)An , n is greater than or equal to 2 The value of A786 ?
Answer is 1/1572

Rishabh Mittal · Accepted Answer

Dear Student ,
Please find below the solution to the asked query :

A1+2A2+3A3+ +n-1 An-1= n2 An   
1A1+2A2+3A3+ +n-2 An-2= n-12 An-1   
2Subtracting 2 from 1 , we getn-1 An-1=n2 An-n-12 An-1n-1 An-1+n-12 An-1=n2 Ann An-1 - An-1 +n2 An-1 -2n An-1+An-1=n2 Ann2 An-1 - n An-1 = n2 AnAnAn-1=n2-nn2AnAn-1=n-1n1n-1 An-1=1n An1n An-1n-1 An-1=1So , it is forming an H
P  1A1 , 12 A2 , 13 A3 
So, 786 A786=11A1+786-11786 A786=11+785=1786A786=17862   ANS
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Let An be a sequence defined by A1=1 then A1+ 2A2 + 3A3+ ... + (n-1)A n-1 = (n^2)An , n is greater than or equal to 2. The value of A786 ? Answer is 1/1572