Question

find the sum of all even natural number from 1 to 150​

Answer 1

ANSWER:

• The sum of all even natural numbers from 1 to 150 = 5700

TO FIND:

• The sum of all even natural numbers from 1 to 150.

EXPLANATION:

Series : 2 + 4 + 6 + …………… + 150

Take 2 as common

2( 1 + 2 + 3 + …………… + 75)

$\red{\bigstar{\boxed{\bold{\gray{Sum\ of\ n\ natural\ number = \dfrac{n(n+1)}{ 2}}}}}}$

n = 75

${2(1 + 2 + 3 + \dots\dots + 75) = \dfrac{2[75(75 + 1)]}{ 2}}$

$2(1 + 2 + 3 + \dots\dots + 75) =75\times76$

$2(1 + 2 + 3 + \dots\dots + 75) =5700$

BY USING A.P FORMULA:

Series : 2 + 4 + 6 + …………… + 150

$\blue{\bigstar{\boxed{\large{\gray{\bold{S_n =\dfrac{n}{2}(2a + (n-1)d)}}}}}}$

n = 75

a = 2

d = 2

$S_{75} =\dfrac{75}{2}(2(2)+ (75-1)2)$

$S_{75} =\dfrac{75}{2}(4+ (74)2)$

$S_{75} =\dfrac{75}{2}(4+ 148)$

$S_{75} =\dfrac{75}{2}(152)$

$S_{75} =75\times 76$

2 + 4 + 6 + …………… + 150 = 5700

Hence the sum of all even natural numbers from 1 to 150 = 5700.