Question

Find sum of all odd numbers between 200 and 300

Answer 1

Odd numbers between 200 and 300 are : 201 , 203 , 205 , 207 ........... , 299 .




AP = 201 , 203 , 205 , 207........., 299.


Here,


First term ( a ) = 201


Common difference ( d ) = 2



Tn = 299



a + ( n - 1 ) × d = 299



201 + ( n - 1 ) × 2 = 299



201 + 2n - 2 = 299



2n + 199 = 299



2n = 299 - 199


2n = 100


n = 50.


Hence,

50 odd numbers are between 200 and 300.

Answer 2

The sum of all odd numbers between 200 and 300 is 12,500.

Solution: The series formed when all the odd numbers between 200 and 300 are written is as follows:

201, 203 ,205, 207 and it goes on till 299 which is the last term of the series.

This series clearly forms an arithmetic progression (AP) whose first term is 201 and the common difference (difference between two successive terms) is 2.

Let the first term of the AP be denoted by a, number of terms by n and common difference be denoted by d.

a= 201

d= 2

The last term of an AP containing n terms is given by the formula:

Last term = a+(n-1)d

=> 299 = 201 + 2(n-1)

=> 299-201 = 2n -2

=> 98+2 = 2n

=> 2n = 100

=> n = 100/2 = 50

This AP contains 50 terms.

Sum of AP is given by the formula

$\frac{n}{2} (2a + (n - 1)d)$

$= \frac{50}{2} (2 \times 201 + 2(50 - 1))$

$= \frac{50}{2} (2 \times 201 + 2 \times 49)$

$= 25(402 + 98)$

= 25× 500

$= 12500$

Therefore, sum of this A.P. = 12,500.