Q.10 : There are n garlands such that the first garland contains 5 flowers, 5the 2nd 7 flowers and in general kth garland contains k flowers more than that in (k-1)th garland. Find the total number of flowers in all n garlands.

Solution :

Let Sn denote total no. of flowers in all n garlands.

Sn = 5 + (2 + 5) + (3 + 2 + 5) + (4 + 3 + 2 + 5) + . Tn

Sn = 5 + (2 + 5) + (3 + 2 + 5) + + Tn-1 + Tn

      Sn Sn = 0 = 5 + {(2+5) 5} + {(3+2+5(-(2+5)} + (Tn - Tn-1) - Tn

= 5 + 2 + 3 + 4 + + n - Tn

      Tn = 5 + 2 + 3 + 4 + .. + n

= 4 + (1 + 2 + 3 + + n)

n(n+1)

= 4 +

2

n r n

Sn = ∑ Tn = ∑ 4 + n2 + ∑ n

r=0 r=1 r=0

n(n+1)(2n+1) n(n+1)

Sn = 4 n + + . (1)

12 4

 

 


... ∑ n2 = 12 + 22 + .. n2 = n(n+1)(2n+1)/6

∑n = 1 + 2 + .. + n = n(n+1)/2

 


So, Sn = n/6 (n2 + 3n + 26) .. [From (1)]

 

So, total number of flowers in all n garlands

 

= Sn = n/6 (n2 + 3n + 26) Ans