Without addition, find : $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19$

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Solution

$\sum_{1}^{n} (2 n + 1) = n^{2}$
Sum of $n$ odd numbers is= $n^{2}$
$1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = (10)^{2} = 100$ (as there are 10 odd numbers n=10)

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1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + . . . . + 33

1 + 3 + 5 + 7 + 9

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23

1 + 3 + 5 + 7 + 9

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23

Without adding find the sum

(i) 1 + 3 + 5 + 7 + 9

(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23

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