Arithmetic Series help (AS Level)

In summary, the problem involves finding the value of a constant, P, in an arithmetic sequence and using it to solve for various terms and the sum of the first 15 terms. To solve part (d), one can try different values of n in the general term formula to find the greatest term less than 400.
  • #1
CathyLou
173
1
I'm totally stuck on the following question and so I'd very very grateful if someone could please tell me how to work it out.

The first three terms of an arithmetic series are (12-P), 2P and (4P-5) respectively, where P is a constant.

(a) Find the value of P.

(b) Show that the sixth term of the series is 50.

(c) Find the sum of the first 15 terms of the series.

(d) Find how many terms of the series have a value of less than 400.

Thank you.

Cathy
 
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  • #2
You have to show some work in order to get help.
 
  • #3
Present us some of your work. Write down the expression for the general term of an arithmetic sequence, and everything should be more clear. Set up a few equations, and see where they'll bring you.

Edit: late again. :smile:
 
  • #4
Before I posted I had written the following in response to part a:

an = a1 + (n - 1)d

d = 2p - (12 - p)

d= (4p - 5) - 2p

2p - (12 - p) = (4p - 5) - 2p

4p = (4p - 5) + (12 - p)

Then I got d = 7 but I got confused over the value of p.

Am I working on the right lines?

Thanks for replying by the way! :smile:

Cathy
 
  • #5
No, p should equal 7.
 
  • #6
radou said:
No, p should equal 7.

Oh, okay, I get where I went wrong. Thank you.

So, for part b do I just substitute in p to the equation to find d?

Cathy
 
  • #7
CathyLou said:
Oh, okay, I get where I went wrong. Thank you.

So, for part b do I just substitute in p to the equation to find d?

Cathy

Yes, you do. I.e., d must equal a3 - a2, and a2 - a1, doesn't matter which difference you take.
 
  • #8
Thanks for your help.

I've now worked out parts a, b and c but I'm still unsure how to do d. Any hints would be really appreciated.

Cathy
 
  • #9
Basically, you can solve part (d) unformally by trying to plug in different numbers n into an = a1 + (n - 1)*d and see which one is the greatest term which is less than 400.
 
  • #10
radou said:
Basically, you can solve part (d) unformally by trying to plug in different numbers n into an = a1 + (n - 1)*d and see which one is the greatest term which is less than 400.

Okay, thanks very much! :biggrin:

Cathy
 

Related to Arithmetic Series help (AS Level)

1. What is an arithmetic series?

An arithmetic series is a sequence of numbers where the difference between any two consecutive terms is constant. The series can be written as a sum of the terms, with the first term being denoted as 'a' and the common difference as 'd'. The general form of an arithmetic series is a, a + d, a + 2d, a + 3d, ... , a + nd, where n is the number of terms in the series.

2. How do you find the sum of an arithmetic series?

The sum of an arithmetic series can be found using the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, 'a' is the first term, and 'd' is the common difference. This formula can also be written as Sn = (n/2)(a + l), where 'l' is the last term in the series. Alternatively, you can also use the formula Sn = (n/2)(a + a + (n-1)d) to find the sum.

3. What is the difference between an arithmetic series and an arithmetic progression?

An arithmetic series is a sum of the terms in an arithmetic progression, while an arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. In other words, an arithmetic progression is a pattern of numbers, while an arithmetic series is the sum of those numbers.

4. How can you tell if a given sequence is an arithmetic series?

To determine if a sequence is an arithmetic series, you can look for two things: a constant difference between consecutive terms, and a pattern of addition or subtraction between terms. If the difference between consecutive terms is the same and the pattern follows addition or subtraction, then the sequence is an arithmetic series.

5. What is the use of arithmetic series in real life?

Arithmetic series have several applications in real life, including compound interest calculations, mortgage payments, and depreciation of assets. They are also used in physics and engineering to model the motion of an object with constant acceleration. Additionally, arithmetic series are used in computer science and coding algorithms.

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