- #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?
Proving the Sum of an Arithmetic Series Formula
- Thread starter yourmom98
- Start date
-
- Tags
- Arithmetic Series
Click For Summary
To verify the formula Sn = n/2[2a + (n-1)d] for the sum of an arithmetic series, one can use a constructive proof by expressing the sum S(n) as both the forward and reverse sequences of terms. By pairing terms from both ends of the sequence, it becomes clear that each pair sums to the same value, leading to the conclusion that the total sum can be simplified. Additionally, using mathematical induction can further validate the formula by demonstrating its correctness through specific examples and applying known summation techniques. Ultimately, the values of a and d are not necessary for the proof, as the structure of the formula holds universally for any arithmetic series. Understanding and applying these concepts requires careful consideration and practice.
- #2
matt grime
Science Advisor
Homework Helper
- 9,361
- 6
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.
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
HallsofIvy
Science Advisor
Homework Helper
- 42,895
- 984
I assume you mean the arithmetic series \sum_{i=1}^n(a+ (i-1)d).
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.
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
yourmom98
- 42
- 0
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
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
matt grime
Science Advisor
Homework Helper
- 9,361
- 6
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.
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
yourmom98
- 42
- 0
Thanks matt i get it now i was a bit confused 
by your 1st post and what you meant thanks for your help
by your 1st post and what you meant thanks for your helpSimilar threads
- · Replies 2 ·
High School
Arithmetic mean of an infinite number of points?
- · Replies 20 ·
- · Replies 3 ·
- · Replies 1 ·
- · Replies 5 ·
- · Replies 7 ·
- · Replies 11 ·
- · Replies 1 ·
- · Replies 5 ·
- · Replies 8 ·