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 — the number you multiply one term by to get the next term in the sequence, where 𝑟 is not equal to one. Identify 𝑎 and 𝑟 in the following sequence. 250, 50, 10, two, and so on.
We are told in the question that the first term of any geometric sequence is 𝑎. This means that in our sequence 𝑎 is equal to 250. The second term 𝑎𝑟 is equal to 50. This means that we can calculate the common ratio 𝑟 by dividing the second term 50 by the first term 250. We can simplify this by dividing the numerator and denominator by 10, leaving us with five over 25. By dividing the numerator and denominator by five, this simplifies further to one-fifth.
In the sequence 250, 50, 10, two, and so on, the first term 𝑎 is equal to 250 and the common ratio 𝑟 is one-fifth. We can check this value of 𝑟 is correct by multiplying the second term by one-fifth to get the third term. 50 multiplied by one-fifth is equal to 10. It is also true that when we multiply the third term 10 by one-fifth, we get the fourth term, which is two.
