Proving the Difference of Sums in an Arithmetic Progression

In summary, the conversation discusses an arithmetic progression with n terms and a common difference of d. The task is to prove that the difference between the sum of the last k terms and the sum of the first k terms is equal to | (n-k)kd |. The equations {S_n} = \frac{n}{2}\left[ {2{a_1} + \left( {n - 1} \right)d} \right] and {u_n} = {a_1} + \left( {n - 1} \right)d are provided for use. The conversation also includes a hint to use the values of a1 and a95 to solve the problem.
  • #1
fluppocinonys
19
1

Homework Statement


An arithmetic progression has n terms and a common difference of d. Prove that the difference between the sum of the last k terms and the sum of the first k terms is | (n-k)kd |.

Homework Equations



[tex]\begin{array}{l}
{S_n} = \frac{n}{2}\left[ {2{a_1} + \left( {n - 1} \right)d} \right] \\
{u_n} = {a_1} + \left( {n - 1} \right)d \\
\end{array}[/tex]


The Attempt at a Solution


I have no idea how to apply the "first 3 terms" and "last 3 terms" into the equation...
Do I use [tex]{u_n}[/tex] as last term, and subsequently [tex]{u_{n - 1}}[/tex], [tex]{u_{n - 2}}[/tex] for last second and third term?
 
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  • #2
Hi fluppocinonys! :wink:
fluppocinonys said:
Do I use [tex]{u_n}[/tex] as last term, and subsequently [tex]{u_{n - 1}}[/tex], [tex]{u_{n - 2}}[/tex] for last second and third term?

Yes. :smile:
 
  • #3
I tried but still unable to solve it.
Can you please hint me on how to start the question with?
thanks
 
  • #4
fluppocinonys said:
I tried but still unable to solve it.
Can you please hint me on how to start the question with?
thanks

Hint: if n = 100, and k = 6, what is the difference between a1 and a95? :smile:
 
  • #5
a95 = a1 + 94d
so,
a95 - a1
= a1 + 94d - a1
= 94d
 
  • #6
fluppocinonys said:
a95 = a1 + 94d
so,
a95 - a1
= a1 + 94d - a1
= 94d

Yup! :biggrin:

and then you … ? :wink:
 

What is an Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is also known as the common difference.

What is the formula for finding the nth term of an Arithmetic Progression?

The formula for finding the nth term of an AP is: a + (n-1)d, where a is the first term and d is the common difference.

What is the sum of the first n terms of an Arithmetic Progression?

The sum of the first n terms of an AP can be calculated using the formula: Sn = (n/2)(2a + (n-1)d), where Sn is the sum, a is the first term, and d is the common difference.

What is the difference between Arithmetic Progression and Geometric Progression?

The main difference between Arithmetic Progression and Geometric Progression is that in AP, the difference between any two consecutive terms is constant, while in GP, the ratio between any two consecutive terms is constant.

What are some real-life examples of Arithmetic Progression?

Arithmetic Progression can be observed in various scenarios, such as the increase in salary every year, the number of stairs in a building, and the distance traveled by a vehicle in equal time intervals.

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