Calculate in head, the sums of the following arithmetic sequences.

i) 51 + 52 + 53 + … + 70

ii)

iii)

(i) Here, we will use formula in which we can calculate sum of AP by first term(a) and last term(a_n) i.e.

Where n is no of terms

Clearly first term, a = 51

Last term, a_n = 70

And no of terms, n = 70 - 51 + 1 = 20

Hence, sum is

= 10(121)

= 1210

(ii)

Here, we will use formula in which we can calculate sum of AP by first term(a) and last term(a_n) i.e.

Where n is no of terms

Clearly first term,

Last term,

Common difference,

Therefore, we can calculate no of terms usinh,

⇒ 11 = n - 1

⇒ n = 12

Hence, we have sum as

⇒ S_n = 6(14) = 84

(iii)

Here, we will use formula in which we can calculate sum of AP by first term(a) and last term(a_n) i.e.

Where n is no of terms

Clearly first term,

Last term,

Common difference,

Therefore, we can calculate no of terms usinh,

⇒ n = 25

Hence, we have sum as