Question

13. Without doing the actual addition, find the sum
of :
(i) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
+ 21 + 23
(ii) 1 + 3 + 5 + 7 + 9 + ................ +39 + 41
(iii) 1 + 3 + 5+ 7 + 9 + ......... +51 + 53
​

Answer 1

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there are 12 numbers which are consecutive odd...therefore we can express them as the Square of total numbers given ...

(i) 12^² = 144

(ii) 21^²= 441

(iii) 27^² = 729

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Answer 2

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

+ 21 + 23

Using Arithmetic Progression

First term (a) = 1

Common difference (d) = 3 - 1 = 2

Last term (l) = 23

Number of terms (n) = 12

We have to find the summative (sum) of the given A.P.

Sn = n/2 (a + l)

→ 12/2 (1 + 23)

→ 6 (24)

→ 144

ii) 1 + 3 + 5 + 7 + 9 + ................ +39 + 41

First term (a) = 1

Common difference = 3 - 1 = 2

Last term (an or l) = 41

an = a + (n - 1)d

OR

l = a + (n - 1)d

41 = 1 + (n - 1)2

40 = (n - 1)2

20 = n - 1

n = 21

Sn = n/2 (a + l)

→ 21/2 (1 + 41)

→ 21/2 (42)

→ 21(21)

→ 441

iii) 1 + 3 + 5+ 7 + 9 + ......... +51 + 53

First term (a) = 1

Common difference = 3 - 1 = 2

Last term (an or l) = 53

an = a + (n - 1)d

OR

l = a + (n - 1)d

53 = 1 + (n - 1)2

52 = (n - 1)2

26 = n - 1

27 = n

Sn = n/2 (a + l)

→ 27/2 (1 + 53)

→ 27/2 (54)

→ 27(27)

→ 729