Question

Study the following pattern:

1 = $\frac{1\times 2}{2}$
1 + 2 = $\frac{2\times 3}{2}$
1 + 2 + 3 = $\frac{3\times 4}{2}$
1 + 2 + 3 + 4 = $\frac{4\times 5}{2}$
By observing the above pattern, find

(i) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

(ii) 50 + 51 + 52 + ................ + 100

(iii) 2 + 4 + 6 + 8 + 10 + ............... + 100

Answer 1

(i) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = $\frac{10\times 11}{2}$ = 55
(ii) 50 + 51 + 52 + ...+ 100
This can also be written as (1 + 2 + 3 + ...+ 99 + 100) - (1 + 2 + 3 + 4 + ...+ 47 + 49)
Now, (1+ 2 + 3 + ...+ 99 + 100 ) = $\frac{100\times 101}{2}$
and, (1 + 2 + 3 + 4 + ...+ 47 + 49 ) = $\frac{49\times 50}{2}$
So, (50 + 51 + 52 + ...+ 100 ) = $\frac{100\times 101}{2}$ - $\frac{49\times 50}{2}$ = 5050 - 1225 = 3825
(iii) 2 + 4 + 6 + 8 + 10 + ... + 100
This can also be written as 2 × (1 + 2 + 3 + 4 + ...+ 49 + 50)
Now, (1 + 2 + 3 + 4 + ...+ 49 + 50 ) = $\frac{50\times 51}{2}$ = 1275
∴ (2 + 4 + 6 + 8 + 10 + ...+ 100) = 2 × 1275 = 2550