### Video Transcript

Consider the following pattern. Which of the following sequences
represents the number of solid blue triangles in each successive term of the
pattern? Is it (A) two, eight, 26, 80, and
so on? (B) One, three, nine, 27, and so
on. (C) Two, six, 18, 54, and so
on. (D) Two, four, 12, 36, and so
on. Or (E) two, four, eight, 16, and so
on. Which type of sequence is found
when counting the number of solid blue triangles in the above pattern?

In this question, we’re interested
in the number of solid blue triangles in each term. We are given five possible
sequences that represents this. In term 1, it is clear that there
are two blue triangles. This immediately rules out option
(B) as the first term in this sequence is one. In the second term, there are six
blue triangles. This rules out option (A), option
(D), and option (E) as these have a second term of eight, four, and four,
respectively. So far, the first two terms match
those in option (C). In pattern 3, each of the sections
circled has three blue triangles. As there are six of these, this
gives a total of 18 blue triangles. This once again corresponds to the
third term in option (C).

In term 4, each of the sections
circled has nine blue triangles, giving a total of 54. The sequence that represents the
number of solid blue triangles is two, six, 18, 54, and so on. We can therefore conclude that the
correct answer is option (C).

In the second part of this
question, we are asked to work out which type of sequence is shown. This could potentially be an
arithmetic sequence, a geometric sequence, or neither. We recall that an arithmetic
sequence has a common difference between consecutive terms. This is clearly not the case for
this sequence. A geometric sequence has a common
ratio between successive terms. As two multiplied by three is six,
six multiplied by three is 18, and 18 multiplied by three is 54, the sequence two,
six, 18, 54 has a common ratio equal to three. We can therefore conclude that the
type of sequence that is found when counting the number of solid blue triangles in
the pattern is a geometric sequence.