# Question Video: Identifying a Geometric Sequence Mathematics • 9th Grade

Consider the following pattern. Which of the following sequences represents the number of solid blue triangles in each successive term of the pattern? [A] 2, 8, 26, 80, … [B] 1, 3, 9, 27, … [C] 2, 6, 18, 54, … [D] 2, 4, 12, 36, … [E] 2, 4, 8, 16, … Which type of sequence is found when counting the number of solid blue triangles in the above pattern?

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### 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.