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Question

Find the sum of the following numbers without actually adding the numbers.
(i) 1+3+5+7+9+11+13+15
(ii) 1+3+5+7
(iii) 1+3+5+7+9+11+13+15+17

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Solution

Given S=1+3+5+7+....+(2n1)
This is an arithmetic progression
Sum of AP: Sn=n2[2a+(n1)d]=n2[2+(n1)2]=n2

(i) 1+3+5+7+9+11+13+15
n=8Sn=n2=82=64

(ii) 1+3+5+7
n=4Sn=n2=42=16

(iii) 1+3+5+7+9+11+13+15+17
n=9Sn=n2=92=81

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