Video Transcript
Find the first five terms of the sequence 𝑎 sub 𝑛, given 𝑎 sub 𝑛 plus one is equal to a quarter multiplied by 𝑎 sub 𝑛, where 𝑛 is greater than or equal to one and 𝑎 sub one is equal to negative 27.
This is an example of a geometric sequence as we are multiplying each term by a common ratio to get the next term. The first term of this geometric sequence, 𝑎 sub one, is equal to negative 27. As we need to multiply this by one-quarter to get the next term, the common ratio 𝑟 is equal to one-quarter. This means that we can calculate the second term, 𝑎 sub two, by multiplying one-quarter by negative 27. This is equal to negative 27 over four. The third term, 𝑎 sub three, is equal to one-quarter multiplied by negative twenty-seven quarters or negative 27 over four. We need to multiply the second term by one-quarter, giving us negative 27 over 16.
To calculate the fourth term, 𝑎 sub four, we multiply this value by one-quarter. This gives us negative 27 over 64. Finally, the fifth term, 𝑎 sub five, is equal to one-quarter multiplied by negative 27 over 64. This is equal to negative 27 over 256. The first five terms of the sequence 𝑎 sub 𝑛 given 𝑎 sub 𝑛 plus one is equal to a quarter of 𝑎 sub 𝑛 and 𝑎 one is equal to 27 are negative 27, negative 27 over four, negative 27 over 16, negative 27 over 64, and finally negative 27 over 256.
