Question

The sequence a1 = 6, an = 3an-1 can also be written as

Answer 1

The sequence with a1 = 6 and an = 3an-1 is a geometric progression with a common ratio of 3, in which each term is three times the previous term.

Step-by-step explanation:

The student has provided the first term of a sequence a1 = 6, and a recursive formula to find any term in the sequence, which is an = 3an-1. What this means is that each term in the sequence is three times the previous term. For example, to find the second term (a2), we would multiply the first term by 3, thus a2 = 3 * 6 = 18. The sequence is therefore part of a geometric series with a common ratio of 3.

Since the student has mentioned sequence expansion and the binomial theorem, these topics might have been discussed in class, but they are not directly relevant to this specific sequence given, which is a simple geometric progression. However, understanding the binomial theorem is important for recognizing different series expansions and the general algebraic manipulation of sequences and series.

Answer 2

The question as presented is incomplete, here is the complete question with the multiple choice:

The sequence a1 = 6, an = 3an − 1 can also be written as:

1) an = 6 ⋅ 3^n
2) an = 6 ⋅ 3^(n + 1)
3) an = 2 ⋅ 3^n
4) an = 2 ⋅ 3^(n + 1)

The correct choice is option 3) an = 2⋅3^n.

If we look at the initial sequence an = 3⋅an-1, and

a1 = 3⋅a0 = 6
a0 = 6/3
a0 = 2

We can now look at the sequence.

a0 = 2
a1 = 6
a2 = 18
a3 = 54
etc...

A common factor in each of those numbers is 2, so we can rewrite the sequence by factoring out 2.

a0 = 2⋅1
a1 = 2⋅3
a2 = 2⋅9
a3 = 2⋅27

The numbers being multiplied by 2 are all factors of 3. So we can rewrite the sequence again as:

a0 = 2⋅3^0
a1 = 2⋅3^1
a2 = 2⋅3^2
a3 = 2⋅3^3

This sequence can now be rewritten as an = 2⋅3^n.