Question

Find the sum of all odd numbers between 100 and 300

Answer 1

To find :

The sum of all the odd numbers between 100 and 300

We know that :

odd numbers between 100 and 300 are:

101,103,105,107,109,111, . . . , . . . ,299

This sequence is forming an AP, whose :

• common difference = 2
• starting term = 101
• ending term = 299
• number of term = ???

Formula to be applied :

$\large \tt{a_n = a+(n-1)d }$

where,

$\tt{a_n = the\: ending\: term }$

n = number of term

d = common difference

a = starting term

Applying the above rule :

$\tt{a_n = a+(n-1)d}$

$\tt{\implies\: 299 = 101+(n-1)2}$

$\tt{\implies \: 299 = 101+2n-2}$

$\tt{\implies \: 2n-2 = 299-101}$

$\tt{\implies \: 2n = 198+2 }$

$\tt{\implies \: n = \frac{\cancel{200}}{\cancel{2}}}$

$\tt{\implies \: n = 100 }$

hence, total number of terms in the sequence = 100

Formula for finding sum :

$\tt{s_n = \frac{n}{2} [2a+(n-1)d]}$

Applying formula :

to find the sum of sequence , we will use the formula :

$\tt{s_n = \frac{n}{2} [2a+(n-1)d]}$

$\tt{\implies \: s_n =\frac{\cancel{100}}{\cancel{2}} [2(101)+(100-1)2]}$

$\tt{\implies \: s_n = 50[202+198]}$

$\tt{\implies \: s_n = 50(400)}$

$\red{\underline{\boxed{\tt{\implies \: s_n = 20,000 }}}}$

hence, the sum of sequence = 20,000

hence, sum of all odd number between 100 to 300 = 20,000

Answer 2

Given:

A list of numbers between 100 and 300.

To Find:

The sum of all the odd numbers between 100 and 300 is?

Solution:

The given problem can be solved using the concepts of Arithmetic Progression.

1. The first odd number after 100 is 101 and the last odd number below 300 is 299.

2. The A.P will be 101, 103, 105,.., 299.

=> a = first term = 101,

=> d = common difference = 2,

=> Tn = last term = 299.

3. The nth term of an A.P with first term a, Common difference d is given by the formula,

=>$T_{n} = a + (n-1)d$.

4. Using the above formula the number of terms can be found,

=> 299 = 101 + (n-1)2,

=> 198 = (n-1)2,

=> 99 = (n-1),

=> n = 100.

5. The sum of the first n terms of an A.P is given by the formula,

=>$S_{n} = \frac{n}{2}(2a + (n-1)d)$.

6. Using the above formula, the sum of the first 100 terms can be found,

=> Sum = (100/2)x(2(101) + (100-1)2),

=> Sum = 50(202+ 198),

=> Sum = 50(400),

=> Sum = 20000.

Therefore, the sum of all the odd numbers between 100 and 300 is 20000.