Question

find the sum of 11 no. staring from 2 to 50​

Answer 1

Answer:

The sum from 1 to n (where n is a positive integer) is ALWAYS given by the formula :

n(n+1)2

and therefore the answer is

50⋅512=25⋅51=1275

But what is the proof that this works.

To prove this - lets assume that this is correct, and prove that it works for a specific known value of n (n=1) and then prove that if it applies for an arbitary value of n it will also apply for a value of n+1. This is proof by induction.

Lets define Sum from 1 to n as S(n)

This is also the same as saying

∑x=nx=1x=S(n)

We can state as fact that :

S(1)=1

S(2)=3

and

S(n)=S(n−1)+n

S(n+1)=S(n)+n+1

Lets assume that S(n)=n(n+1)2

We can show that under that definition:

S(1)=1(1+1)2=22=1

so our definition works for n=1

So what is S(n+1)

We expect it to be

S(n+1)=(n+1)(n+2)2=S(m)=m(m+1)2

But we also expect this relationship to be true :

S(n+1)=S(n)+n+1

So does this work with our definition of S(n)?

S(n+1)=n(n+1)2+(n+1)=

n(n+1)2+2(n+1)2=n(n+1)+2n+22=n2+n+2n+22=n2+3n+22=(n+1)(n+2)2=

So we know that

S(n)=n(n+1)2 is right when n=1

and we know that if S(n)=n(n+1)2 is right for n then it must be right for n+1