Question regarding summation of series

In summary, the conversation discusses the formula for finding the sum of an arithmetic progression and how to express it in a specific way. The solution involves substituting values and expanding expressions to find the general term, which can then be used to find the sum up to a specific term.
  • #1
misogynisticfeminist
370
0
If [tex]\sum^{n}_{r=1} u_r =3n^2 +4n [/tex], what is [tex] \sum^{n-1}_{r=1}u_r [/tex] ?

I know that [tex] \sum^{n-1}_{r=1}u_r [/tex] is equals to [tex]\sum^{n}_{r=1} u_r =3n^2 +4n - u_n [/tex] but the answer given is [tex] 3n^2-2n-1[/tex]. How do i express it in that way?

thanks alot.
 
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  • #2
Use this

[tex] \sum_{r=1}^{n} r =\frac{n(n+1)}{2} [/tex]

Daniel.
 
  • #3
misogynisticfeminist said:
If [tex]\sum^{n}_{r=1} u_r =3n^2 +4n [/tex], what is [tex] \sum^{n-1}_{r=1}u_r [/tex] ?

I know that [tex] \sum^{n-1}_{r=1}u_r [/tex] is equals to [tex]\sum^{n}_{r=1} u_r =3n^2 +4n - u_n [/tex] but the answer given is [tex] 3n^2-2n-1[/tex]. How do i express it in that way?

thanks alot.
Substitute n with n-1:

[tex]\sum^{n}_{r=1} u_r =3n^2 +4n [/tex]
becomes
[tex]\sum^{n-1}_{r=1} u_r =3(n-1)^2 +4(n-1)
=3(n^2 -2n +1) +4(n -1)
=3n^2 -2n -1[/tex]
 
  • #4
Heh,again,thank god someone came with a more intuitive method...:smile:

I would have found the general term [itex] u_{r}=6r+1 [/itex] ...

Daniel.
 
  • #5
dextercioby said:
Use this

[tex] \sum_{r=1}^{n} r =\frac{n(n+1)}{2} [/tex]

Daniel.

hey thanks daniel. I've used that to find out that [tex] sum^{n}_{r=1} u_r [/tex] is an arithmetic progression with difference 6 and first term 7. So, i went on from there to find out the answer.

to whkoh: i don't understand your second step, [tex]\sum^{n-1}_{r=1} u_r =3(n-1)^2 +4(n-1)
=3(n^2 -2n +1) +4(n -1)
=3n^2 -2n -1[/tex].

would you mind explaining it to me? thanks alot.
 
  • #6
The second step is expanding out so that it becomes
[tex]3(n-1)^2+4(n-1)
=3(n^2 -2n+1)+4n-4
=3n^2 -6n+4n+3-4
=3n^2 -2n-1
[/tex]
 
  • #7
I don't understand how you find the general term [tex]U_r[/tex]?
 
  • #8
[tex] 6 \sum_{r=1}^{n} r =6\frac{n(n+1)}{2}=3n^{2}+3n [/tex] (1)

U have [tex] 3n^{2}+4n [/tex],so you need another "n"...That can be gotten noticing that

[tex] \sum_{r=1}^{n} 1 = n [/tex] (2)

Add (1) & (2) and u'll get

[tex] \sum_{r=1}^{n} \left(6r+1\right) = 3n^{2}+4n [/tex] (3)

The conclusion is simple.U found the general term,so for a summation till [itex] n-1 [/itex] simply subtract the general term from the sum till [itex] n [/itex]...

Daniel.

P.S.Your method is simpler,mine is due to intuition only.
 
  • #9
ahhh thanks alot. I finally understood..
 

Related to Question regarding summation of series

What is the purpose of summation of series?

The purpose of summation of series is to find the total value of a sequence of numbers by adding them together. It is often used in mathematics, statistics, and science to calculate the sum of a large set of data or to find patterns in a series.

What is the formula for calculating summation of series?

The formula for calculating summation of series is Σn = a1 + a2 + a3 + ... + an, where n is the number of terms in the series and ai represents each individual term in the series.

What are some common types of series that are summed?

Some common types of series that are summed include arithmetic sequences (where each term is obtained by adding a constant number to the previous term), geometric sequences (where each term is obtained by multiplying a constant number to the previous term), and power series (where each term is obtained by raising a constant number to a power).

What is the difference between finite and infinite series?

A finite series has a fixed number of terms, while an infinite series continues indefinitely. In other words, a finite series has a last term, while an infinite series does not. This means that the sum of a finite series will eventually reach a final value, while the sum of an infinite series may continue to approach a certain value without ever reaching it.

How is summation of series used in real life?

Summation of series has many practical applications in everyday life. For example, it can be used to calculate total expenses or income over a period of time, to find the average value of a set of data, or to predict future trends based on past data. It is also used in fields such as physics, engineering, and economics to model real-world phenomena and make predictions based on mathematical patterns.

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