Let a₁ be the first term in the sequence. We can use the fact that a_n+1 = (a_n)² – 1 in order to find expressions for the second and third terms of the sequence in terms of a₁.

a₂ = (a₁)² – 1

a₃ = (a₂)² – 1 = ((a₁)² – 1)² – 1

We can use the fact that, in general, (a – b)² = a² – 2ab + b² in order to simplify the expression for a₃.

a₃ = ((a₁)² – 1)² – 1

= (a₁)⁴ – 2(a₁)² + 1 – 1 = (a₁)⁴ – 2(a₁)²

We are told that the third term is equal to the square of the first term. 

a₃ = (a₁)²

We can substitute (a₁)⁴ – 2(a₁)² for a₃.

(a₁)⁴ – 2(a₁)² = (a₁)²

Subtract (a₁)² from both sides.

(a₁)⁴ – 3(a₁)² = 0

Factor out (a₁)² from both terms.

(a₁)² ((a₁)² – 3) = 0

This means that either (a₁)² = 0, or (a₁)² – 3 = 0. 

If (a₁)² = 0, then a₁ must be 0. However, we are told that all the terms of the sequence are positive. Therefore, the first term can't be 0.

Next, let's solve (a₁)² – 3 = 0.

Add 3 to both sides.

(a₁)² = 3

Take the square root of both sides. 

a₁ = ±√3

However, since all the terms are positive, the only possible value for a₁ is √3.

Now, that we know that a₁ = √3, we can find a₂, a₃, and a₄.

a₂ = (a₁)² – 1 = (√3)² – 1 = 3 – 1 = 2

a₃ = (a₂)² – 1 = 2² – 1 = 4 – 1 = 3

a₄ = (a₃)² – 1 = 3² – 1 = 9 – 1 = 8

The question ultimately asks for the product of the a₂, a₃, and a₄, which would be equal to 2(3)(8), or 48.

The answer is 48.

The fourth term is: 3(23) – 1 = 69 – 1 = 68.

The fifth term is: 3(68) – 1 = 204 – 1 = 203.

The sixth term is: 3(203) – 1 = 609 – 1 = 608.

Each number in the sequence in 7 more than the number preceding it.

The equation for the terms in an arithmetic sequence is a_n = a₁ + d(n-1), where d is the difference.

The formula for the terms in this sequence is therefore a_n = 2 + 7(n-1). 

Plug in 8 for n to find the 8^th term: 

a₈ = 2 + 7(8-1) = 51

The difference between the second and fourth terms of an arithmetic sequence is twice the common difference - or, equivalently, the common difference is half the difference between the second and fourth terms.

The common difference can be subtracted from the second term to obtain the first term:

Take the equation that represents the th term in the sequence and plug in the value of 9 for :

The value of the 9th term is 747.

From the given information, we know , which means each consecutive difference is 3.

Each term in the sequence is one less than twice the previous term.

So,  

The series is defined by n² – 1 starting at n = 1. The sixth number in the series then equal to 6² – 1 = 35.

The pattern of the sequence is (x+1) * 2.

We have the first 5 terms, so we need terms 6 and 7:

(78+1) * 2 = 158

(158+1) * 2 = 318

3 + 8 + 18 +38 + 78 + 158 + 318 = 621

Notice that in the sequence 

each term increases by .

It is always good strategy when attempting to find a pattern in a sequence to examine the difference between each term.

We continue the pattern to find:

The  term is 

The  term is 

The  term is 

It is useful to note that the sequence is defined by,

where n is the number of any one term.

We can solve

to find the  term.