Arithmetic progression question

[Michael_Light]

Homework Statement

Series Q is an arithmetic series such that the sum of its first n even terms is more than the sum of its first n odd terms by 4n. Find the common difference of the series Q. The answer provided is 4.

Homework Equations

The Attempt at a Solution

I have no ideas on this... Can you help me?

 

[LCKurtz]

You don't mean the sum of the even terms, you mean the even numbered terms. The sum of the first n even numbered would be:

a₂+a₄+...a_2n

and the odd would be

a₁+a₃+...a_2n-1

Write both sums in terms of a₁ and the unknown d and use your given equation that one is 4n larger than the other.

 

[Michael_Light]

You don't mean the sum of the even terms, you mean the even numbered terms. The sum of the first n even numbered would be:

a₂+a₄+...a_2n

and the odd would be

a₁+a₃+...a_2n-1

Write both sums in terms of a₁ and the unknown d and use your given equation that one is 4n larger than the other.

I have difficulty on forming an appropriate equation to find the common difference... can you help me..?

 

[eumyang]

I found it easier to define two new sequences, b, and c,
where b contains the odd-numbered terms of sequence a, and
where c contains the even-numbered terms of sequence a.

So, for sequence b:
b1 = a1
b2 = a3 = a1 + 2d
b3 = a5 = a1 + 4d
bn = a(2n-1) = ?

Then find the sum of the first n terms of sequence b. Repeat the process for sequence c. Then plug into
(sum of n terms in seq. b) + 4n = (sum of n terms in seq. c)
and solve for d.

 

[HallsofIvy]

If the common difference between terms in the original series is d, then the series of odd indexed terms and the series of even indexed terms are arithmetic series with common difference 2d.