Question

Find the Sum of all integers even integers 2 to 101​

Answer 1

Answer :

$\large{\dag\:\boxed{\star\:\:a_{50}\:=\:2550\:\:\star}\:\dag}$

Explanation :

Given :–

• A.P.  :- 2 , 4 , 6 , 8 , ... , 100 .
• Where a = 2 , d = 2 and aₙ = 100 .

To Find :–

• Sum of all the terms of the Above A.P.

Formulae Applied :–

• $\boxed{\bf{\star\:\:a_n\:=\:a\:+\:(n\:-\:1)d\:\:\star}}$

• $\boxed{\bf{\star\:\:S_n\:=\:\dfrac{n}{2}\:[a\:+\:a_n(l)] \:\:\star}}$

Solution :–

First we will find the number of terms (n) :

We have , a = 2 , d = 2 and aₙ = 100 .

$\rightarrow\sf{100\:=\:2\:+\:(n\:-\:1)(2)}$

$\rightarrow\sf{100\:=\:2\:+\:2n\:-\:2}$

$\rightarrow\sf{100\:=\:2n}$

$\rightarrow\sf{n\:=\:\dfrac{100}{2} }$

$\rightarrow\bf{n\:=\: 50 }$

Now , we have a = 2 , aₙ(l = last term) = 100 and n = 50 .

Putting these values in the Formula :

$\rightarrow\sf{S_n\:=\:\dfrac{n}{2}\:[a\:+\:a_n]}$

$\rightarrow\sf{S_{50}\:=\:\dfrac{50}{2}\:\times\:[2\:+\:100]}$

$\rightarrow\sf{S_{50}\:=\:25\:\times\:102}$

$\rightarrow\boxed{\bf{S_{50}\:=\:2550}}$

∴ The Sum of all even integers between 2 to 101 is 2550 .