# The sum of all odd #'s between 2 and 30.

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Odd numbers between 2 and 30 would be the sequence:

3, 5, 7, ...., 29

The sum of odd numbers from 1 to n, where n is an odd number, formula is:

1 +3 + ....+ 2n-1 = n^2

then,

1 +3+....+ 2(15)-1 = (15)^2

by subtracting 1 from both sides:

3+5+...+ 29 = (15)^2 - 1 = 225-1=224

the sum of the odd numbers between 2 and 30 equals 224

The sum of all the odd numbers between 2 & 30 is :-

3+5+7+9+11+13+15+17+19+21+23+25+27+29=224

It has some easy methods but it will take a lots space to write.Thatswhy i have not written the easy methods.

The given series is:

3 + 5 + 7 ... + 27 + 29

This can be represented as:

(2x1 + 1) + (2x2 + 1) + (2x2 + 1) ... + (2x13 + 1) + (2x14 + 1)

These can be separated in to two series as follows:

[2x1 + 2x2 + 2x3 ... 2x13 + 2x14] +

[1 + 1 + 1 ... + 1 + 1 ]

= 2x(1 + 2 + 3 ... + 13 + 14) + 14

= 2x[(14x15)/2] + 14

= 2x105 + 14 = 210 + 14 = 224

Sum of the odd numbers btween 2 to 30 is equal to:

3+5+7+9+......27+29, which is in AP.

The common difference d = 2. The number of terms, n = (last term - 1st term)/d + 1 = (29-3)/2+1 = 13+1 =14.

Therefore,the required sum = (First term + Last term)n/2 = (3+29)*14/2 = 32*14/2 = 224.