Question

(1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,)=?+?+?=30 how find it?

Answer 1

Question doesn't seems to be written properly but I think that you are trying to find how the given list generates sum of 30.

Given sequence {1,3,5,7,9,11,13,15,17,19,21,23,25,27,29} is an Arithmetic sequence. because difference between each consecutive terms is 2 which is constant.

In this type of sequence if we make pairs of first term from start and end, pair of 2nd term from start and end, and so on. then we will get pairs:

(1,29)

(3,27)

(5,25).... etc.

Now if you check the sum of each pair then we will find that sum is always same

1+29=30

3+27=30

5+25=30

That is how you will get sum of 30 for given problem.

This property of arithmetic sequence is found by mathematician Gauss.

Answer 2

1+3+5+...+(2n-1) = ? Let's look at the problem for n = 1,2,3,4,5

1=1=1

2

1+3=4=2

2

1+3+5=9=3

2

1+3+5+7=16=4

2

1+3+5+7+9=25=5

2

So the answer seems to be:

1+3+5+...+(2n-1) = n

2

Step 1.

For n = 1 it's true that 1 = 2*1-1

Step 2.

We suppose that 1+3+5+...+(2n-1) = n^2

and want to prove that: 1+3+5+...+(2(n+1)-1) = (n+1)^2

We add (2(n+1) -1) to this:

1+3+5+...+(2n-1) = n^2

and get:

1+3+5+...+(2n-1) + (2(n+1) -1) = (n^2)+ (2(n+1) -1)

so:

1+3+5+...+(2(n+1) -1) = (n^2)+ 2n+2 -1

but n^2+2n+1 = (n+1)^2

so we finally have:

1+3+5+...+(2(n+1)-1) = (n+1)^2

so here n=50; and answer for this series 1,3,5,7,…..,97,99 will be

n^2= 50^2=2500.