# Arithmetic Sequence

1. Apr 4, 2013

### FeDeX_LaTeX

1. The problem statement, all variables and given/known data

This is taken from STEP I 1990, Q4.

(i) The sequence a1, a2, ..., an, ... forms an arithmetic progression. Establish a formula, involving n, a1, and a2, for the sum of the first n terms.

(ii) A sequence b1, b2, ..., bn, ... is called a double arithmetic progression if the sequence of differences, b2 - b1, b3 - b2, ..., bn+1 - bn, ... is an arithmetic progression. Establish a formula, involving n, b1, b2 and b3, for the sum b1 + b2 + ... + bn of the first n terms of such a progression.

(iii) A sequence c1, c2, ..., cn, ... is called a factorial progression if cn+1 - cn = n!d, for some non-zero d and every n ≥ 1. Suppose 1, b2, b3, ... is a double arithmetic progression, and also that b2, b4, b6 and 220 are the first four terms in a factorial progression. Find the sum 1 + b1 + b2 + ... + bn.

2. Relevant equations

Standard arithmetic progression formulae below

The nth term of an AP: un = a + (n-1)d
The sum of the first n terms of an AP: Sn = (n/2)(a + l) = (n/2)(2a + (n-1)d)

3. The attempt at a solution

I've done (i) quite comfortably and got

$$\frac{n}{2}((3-n)a_{1} + (n-1)a_{2})$$

However, (ii) is where I get stuck. By considering the sequence of differences, I've established that

$$b_n = a + (n-2)d + b_{n-1}$$

with a = b2 - b1, and d = (b3 - b2) - (b2 - b1). Can anyone guide me on where to go next?

Last edited: Apr 4, 2013
2. Apr 4, 2013

### Staff: Mentor

With n=2, I get a1+a2, but the result should be a2.

An explicit formula for bn could be useful. In your formula, you can express bn-1 in terms of bn-2 and so on, until you reach b1.

3. Apr 4, 2013

### FeDeX_LaTeX

Why? We were looking for the sum of the first n terms. With n = 2, that is a1 + a2.

Ah I see, thanks. I will try this and reply if I get the correct result.

4. Apr 4, 2013

### SammyS

Staff Emeritus
Yes, you were correct, FeDeX_LaTeX .

5. Apr 4, 2013

### Staff: Mentor

Oh sorry, I did not see that (a) should be a sum of the first n terms as well.