Question

Study the following pattern:

1 + 3 = 2 × 2
1 + 3 + 5 = 3 × 3
1 + 3 + 5 + 7 = 4 × 4
1 + 3 + 5 + 7 + 9 = 5 × 5

By observing the above pattern, find

(i) 1 + 3 + 5 + 7 + 9 + 11
(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15
(iii) 21 + 23 + 25 + ........ + 51

Answer 1

(i) 1 + 3 + 5 + 7 + 9 + 11 = 6 × 6 = 36
(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 8 × 8 = 64
(iii) 21 + 23 + 25 + ... + 51 =
(21 + 23 + 25 + ... + 51) can also be written as (1 + 3 + 5 + 7 + ...+ 49 + 51) − ( 1 + 3 + 5 + ...+ 17 + 19).
(1 + 3 + 5 + 7 + ...+ 49 + 51) = 26 × 26 = 676
and, (1 + 3 + 5 + ...+ 17 + 19 ) = 10 × 10 =100
Now,
(21 + 23 + 25 + ...+ 51) = 676 − 100 = 576