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

Question

is there any formula to calculate the sum?

Answer 1

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.

Note: If you are a genius you might by chance or investigation note that

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.

So there are many different but equally interesting ways to solve your problem.

Answer 2

let's try to find a pattern:
1st natural odd # = 1
sum of 1st natural #: 1

1st 2 odd nat #s = 1,3
sum of first 2 = 4

1st, 2nd, 3rd: 1,3,5
sum of first 3=9

....1,3,5,7
sum of first 4=16

wait a minute.... i know another pattern that works like that:
1^2=1
2^2=4
3^2=9
4^2=16
The sum of the first X natural numbers is X^2.

Sum of the 1st 25= 25^2
625

Answer 3

It's 25² = 625.
The sum of the first n natural odd numbers is n².
This can be proved by induction.

Answer 4

This Site Might Help You.

RE:
What is the sum of the first 25 natural odd numbers?
is there any formula to calculate the sum?

Answer 5

Is 25 Even Or Odd

Answer 6

1 + 3 + 5 +...+ 49 + 51=
=1 + 2 + 3 + 4 + 5+..+51 - (2 + 4 + 6 + 8 +...+ 50)=
(using gauss formula)
=51*52/2 -2(25*26)/2=
=51*26-25*26=
26*26=676

generally 1+3+5+7+...+(2n+1)=(n+1)^2

Answer 7

n^2