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1. What is the value of 1+3+6+10+15+21+.........+an
Solution
Let S = 1+3+6+10+15+21+.........+an
Rewriting the equation S = 1+3+ 6+10+15+21+ ...........+an
Subtracting the equation (2) from (1)
0 = 1 + 2 + 3 + 4 + 5 + ............... - an
an = 1 + 2 + 3 + 4 + 5 + ....... n terms = n(n+1)/2
[Sum of n natural nos .= n(n+1)/2]
an = (n2 + n)/2
S n = 1/2  (k 2 + k) Sn = 1/2 [n(n+1)(2n+1)/6 + n(n+1)/2]
Sn = n(n+1)/4 [(2n+1)/3 + 1]
Sn = n(n+1)/4 [(2n+4)/3]
Sn = n(n+1)(n+2)/6 Ans
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