Proving the Sum of an Arithmetic Series Formula

In summary, the formula Sn=n/2[2a+(n-1)d] represents the sum of n terms in an arithmetic series. This can be verified by working out a constructive proof using the sum formula S(n)=a(1)+a(2)+...+a(n), or by using induction. The values of a and d do not matter in the proof, and it is important to carefully follow the hints given and think about the solution before it becomes clear.
  • #1
yourmom98
42
0
i am given an formula Sn= n/2[2a+(n-1)d] and i am told to verify the formula represents the sum of n terms of an arithmetic series. How do i verify this?
 
Mathematics news on Phys.org
  • #2
by doing it...

1. work out a constructive proof (if it's any help i believe there's a famous example of this from a child in class one day, asked to count the number of birds if there war 1 on the first step, 2 on the second, 3 on the third etc). hint, let S(n) be the sum of the first n terms of the series

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

well, S(n) also equals

a(n)+...+a(2)+a(1)

and what is a(n)+a(1) and a(n-1)+a(2) and a(n-2)+a(3)...

don't "just do it", apply some high faluting and uniformative mathematics, namely,

2. do it by induction, if you know what that is.
 
  • #3
I assume you mean the arithmetic series [tex]\sum_{i=1}^n(a+ (i-1)d)[/tex].

Start by looking at some examples: if a= 1, d= 3, then the sum is
1+ 4+ 7+ 10= (1+ 0(3))+ (1+ 1(3))+ (1+ 2(3))+ (1+ 3(3)) (here n= 4). I would think of that as (1+ 1+ 1+ 1)+ (0+ 1+ 2+ 3)(3). Obviously the first part of that is just 1 (a in general) added to itself n times: an while the other is 3 (d in general) times the sum of the series 1+ 2+ 3+...+ n-1. Do you know a formula for that. If not use Gauss' idea: Add 1+ 2+ 3+ ...+ (n-1) to (n-1)+...+ 3+ 2+ 1. If you add term by term you get
1+ (n-1)= n, 2+ (n-2)= n, ... up to (n-1)+ 1= n. In other words, every term is n and there are n-1 terms: the sum is n(n-1). Oops! That was adding the sum twice (once in the original order and then reversed) so we need to divide by 2. That sum is n(n-1)/2.
That is, we have a added to itself n times: an and we have d times n(n-1)/2:
an+ 2n(n-1)/2.
 
  • #4
well the whole question that i am asked to do is this

Recall that t1 = a, and tn = a + (n - 1)d for an arithmetic sequence. Verify that the following formula represents the sum of n terms of an arithmetic series:

Sn=n/2[2a+(n-1)d]

i don't know the summation notation yet and i am not given any values for "a" and "d" so i don't know how to actually "do" it
 
  • #5
but the values of a and d do not matter. the "just do it proof" is, relatively straight forward if you follow the hitns you've been given:

a+(a+d)+(a+2d)+...

and

a+(n-1)d+(a+(n-2)d)+...

add up the first and the last terms (which I;ll do for you: it is a and a+(n-1)d), the second and second to last terms (a+d and a+(n-2)d), the third and third to last terms and what do you get? (if the answers are different you'e done something wrong).
did you acutally think about the hints you were given? maths solutions don't suddenyl appear unless you are a genius, you need to think about them for a LONG time before it becomes clear.
 
Last edited:
  • #6
Thanks matt i get it now i was a bit confused :confused: by your 1st post and what you meant thanks for your help
 

1. What is an arithmetic series?

An arithmetic series is a mathematical sequence of numbers where each term is obtained by adding a constant value to the previous term. For example, in the series 2, 5, 8, 11, the constant value is 3.

2. What is the formula for finding the sum of an arithmetic series?

The formula for finding the sum of an arithmetic series is: S = (n/2)(a1 + an), where S is the sum, n is the number of terms, a1 is the first term, and an is the last term.

3. How do you verify if a series is arithmetic?

To verify if a series is arithmetic, you can check if the difference between each consecutive term is constant. If the difference is the same for all terms, then the series is arithmetic.

4. What is the difference between an arithmetic series and a geometric series?

An arithmetic series has a constant difference between each term, while a geometric series has a constant ratio between each term. In other words, in an arithmetic series, the terms increase or decrease by a fixed amount, while in a geometric series, the terms are multiplied or divided by a fixed number.

5. Can an arithmetic series have an infinite number of terms?

Yes, an arithmetic series can have an infinite number of terms as long as the difference between each term remains constant. However, the sum of an infinite arithmetic series can only be calculated if the common difference is less than 1.

Similar threads

Replies
20
Views
1K
Replies
2
Views
993
Replies
1
Views
1K
Replies
3
Views
954
Replies
11
Views
446
Replies
8
Views
865
Replies
5
Views
1K
  • General Math
Replies
1
Views
1K
  • General Math
Replies
7
Views
1K
  • General Math
Replies
5
Views
1K
Back
Top