# Question Video: Finding the First Term and Common Ratio in a Given Geometric Sequence Mathematics

A geometric sequence is a list of terms which can be written in the form 𝑎, 𝑎𝑟, 𝑎𝑟², 𝑎𝑟³, ..., where 𝑎 is the first term and 𝑟 is the common ratio (the number you multiply one term by to get the next term in the sequence, 𝑟 ≠ 1). Identify 𝑎 and 𝑟 in the following sequence: 4, 12, 36, 108, ...

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### Video Transcript

A geometric sequence is a list of terms which can be written in the form 𝑎, 𝑎𝑟, 𝑎𝑟 squared, 𝑎𝑟 cubed, and so on, where 𝑎 is the first term and 𝑟 is the common ratio. This is the number you multiply one term by to get the next term in the sequence, and 𝑟 is not equal to one. Identify 𝑎 and 𝑟 in the following sequence: four, 12, 36, 108, and so on.

We are reminded in the question that 𝑎 is the first term of any geometric sequence. This means that in the sequence four, 12, 36, 108, and so on, 𝑎 must be equal to four. 𝑟 is the common ratio of a geometric sequence. This is the number we multiply the first term by to get the second term. Also, to get from the second to the third term, the third to the fourth term, and so on, we need to multiply by 𝑟.

This means that we can calculate 𝑟 by dividing the second term by the first term. 12 divided by four is equal to three. We can check this by dividing the third term, 36, by the second term, 12. This is also equal to three. Dividing the fourth term 108 by 36, the third term, would also give us an answer of three.

In the sequence four, 12, 36, 108, and so on, the first term 𝑎 is equal to four and the common ratio 𝑟 is equal to three.