Arithmetic Sequence Homework: Find the Sum

[FeDeX_LaTeX]

Homework Statement

This is taken from STEP I 1990, Q4.

(i) The sequence a₁, a₂, ..., a_n, ... 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, ..., b_n, ... is called a double arithmetic progression if the sequence of differences, b₂ - b₁, b₃ - b₂, ..., b_n+1 - b_n, ... is an arithmetic progression. Establish a formula, involving n, b₁, b₂ and b₃, for the sum b₁ + b₂ + ... + b_n of the first n terms of such a progression.

(iii) A sequence c₁, c₂, ..., c_n, ... is called a factorial progression if c_n+1 - c_n = 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 + ... + b_n.

Homework Equations

Standard arithmetic progression formulae below

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

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?

 

[mfb]

I've done (i) quite comfortably and got

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

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

Can anyone guide me on where to go next?

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

 

[FeDeX_LaTeX]

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

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

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

 

[SammyS]

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

Yes, you were correct, FeDeX_LaTeX .

 

[mfb]

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