To find the sum of first n numbers

Achatesopo8

Achatesopo8

Answered question

2022-04-11

To find the sum of first n numbers using the sum of odd numbers and even numbers
We know that the sum of first n numbers is n(n+1)2. So, I tried to derive it using the formula for sum of first n odd numbers and sum of first n even numbers. Here's what I did
What I tried was to split
1+2+3++n
into
(1+3+5++(2n1))+(2+4+6++2n)

Answer & Explanation

davelucasnp0j

davelucasnp0j

Beginner2022-04-12Added 16 answers

Notice that the second part of the expression, (2+4+6++2n) in particular, is equal to twice the sum of the first n numbers. This is because
2+4+6++(2n2)+2n
2(1)+2(2)+2(3)++2(n1)+2(n)
2(1+2+3++(n1)+n)
For you to get the sum of the first n numbers using the formula for odd and even numbers, you have to consider the case where n is odd and n is even. When n is even, then n can be represented as 2k where kN. Then, there must be k even numbers and k odd numbers. Otherwise, there must be k+1 odd numbers and k even numbers.
To avoid confusion, we'll use i as our variable here.
We know that
1+3+5++(2i3)+(2i1)=i2
and 2+4+6++(2i2)+2i=i(i+1).
- Case I: The number of terms is even.
Let n be the number of terms. This means that n=2k where kN. Hence, we have k even numbers and k odd numbers. Getting the sum of k even numbers and k odd numbers,
2+4+6++2k=k(k+1)

1+3+5++2k1=k2
Adding both the sums,
k(k+1)+k2

k2+k+k2

2k2+k

k(2k+1)

n2(n+1)

n(n+1)2
- Case II: The number of terms is even.
To solve this case, we will use the first case, and just add 2k+1. Hence, we have n+1 terms. Then, 
k(k+1)+k2+2k+1

k2+k+k2+2k+1

2k2+3k+1

(2k+1)(k+1)

(n+1)(n2+1)

(n+1)(n+22)

(n+1)(n+2)2
Our last expression seems to match the formula from the first case, only that n became n+1.

Gonarsu2dw8

Gonarsu2dw8

Beginner2022-04-13Added 19 answers

In order to find the proof you are searching for you need to emply a trick. As a hint, note that
(n+1)2n2=2n+1

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?