Question

find the sum of all even Number between 1 and 150

Answer 1

it forms an ap:2,4,6,8,.....150
first term,a=2; common difference,d=2
we know nth term=a+(n-1)d
therefore, 150=a+(n-1)d
150=2+(n-1)2
150=2+2n-2
150=2n
n=75
now, we know,sum of n terms=n/2(2a+(n-1)d)
therfore
sum of 75 terms, with first term,a=2,common difference=2
75/2(2*2+(75-1)2
75/2(4+74(2))
75/2(4+148)
75/2*152
75*76
5700
hence the sum of all even numbers from 1 to 50 is 5700(answer)
thank you;-)

Answer 2

This question can be solved by using concept of Arithmetic Progression (AP) .

The AP is 2 , 4, 6 ,..........150 .


Now, the first even number between 1 to 150=2

So, a1 =2

Last even number between 1 to 150=150

So, an =150

common difference (d)=second term-first term

=4-2

=2

we know that

an=a+(n-1) d

150= 2 + (n-1) ×2

150-2 = (n-1) ×2

148 = (n-1) ×2

148/2 =n-1

74 = n-1

74 +1 = n

75 = n

We know that,

$sn = \frac{n}{2} (a + an)$


$= \frac{75}{2} (2 + 150)$


$= \frac{75}{2} \times 152$


$= \frac{75}{ \cancel{2}} \times \cancel{152} \: 76$


=75×76

= 5700

So, sum of all even number between 1 to 150 is $\bf{5700}$

$\bf{hope \: it \: helps}$