Question

What is the sum of the first 25 natural odd numbers?

Answer 1

Hey dear....

Here is your answer........

From 1 to 50 there are 50 natural numbers, half even and half odd, so we want the sum of the 25 odd numbers from 1 to 49. 

The sum S can be written forwards or backwards as: 

S = 1 + 3 + 5 + ........... + 45 + 47 + 49 or 
S = 49 +47 +45 + ............. + 5 + 3 + 1 

Notice that adding each vertical pair gives 50. 

How many 50s do we get? Looks like 25 of them which add up to 50 x 25 = 1250. But the 50s came from two of the sums S we want so we take half to get S = 625. 

Now can you find the sum of the first 100 odd numbers 
1 + 3 + 5 + ...... + 199 the same way? You should get 10,100. 



1 = 1^2 

1+ 3 = 4 = 2^2 

1+3+5 = 9 = 3^2 

1+3+5+7 = 16 = 4^2 

1+3+5+7+9 = 25 = 5^2 

leading you to guess (and prove later by induction) that 

1 + 3 + 5 + ..... 49 (first 25 odds) = 25^2 = 625. 

Hey dear... Hope it helps you!!!!!

Answer 2

The number series 1, 3, 5, 7, 9, . . . . , 49.

The first term a = 1

The common difference d = 2

Total number of terms n = 25

then apply the input parameter values in the AP formula

Sum = n/2 x (a + Tn)

= 25/2 x (1 + 49)

= (25 x 50)/ 2

= 1250/2

1 + 3 + 5 + 7 + 9 + . . . . + 49 = 625