Question

Find the explicit formula for the sequence 1, -9, -19, ...

a) a_n = 1 - 5n
b) a_n = 1 - 10n
c) a_n = 5 - n
d) a_n = 10 - n

Answer 1

The correct explicit formula for the sequence
`1, -9, -19` is b)
`a_n = 1 - 10n` which accounts for the decrease of
`10` between each consecutive term.

Step-by-step explanation:

To find the explicit formula for the given sequence
`1, -9, -19` we need to observe how the sequence is changing from one term to the next. We can note that the difference between consecutive terms is
`-10` :

• 
  `1 to -9 is a decrease of 10.-9 to -19 is a decrease of 10`.

Therefore the explicit formula will involve subtracting
`10 times` the term's position (n) minus some constant to get the initial term
`1` when

`n = 1`.

Let's check each option:

• 
  `a_n = 1 - 5n`: If we plug in
  `n = 1`, we get
  `-4`, which is not correct.
• 
  `a_n = 1 - 10n`: Plugging in
  `n = 1` gives
  `1 - 10(1) = -9`, which is correct for the second term, so this looks promising.
• 
  `a_n = 5 - n`: Plugging in
  `n = 1` gives
  `4`, which is not correct.
• 
  `a_n = 10 - n`: Plugging in
  `n = 1` gives
  `9`, which is not correct.

So the correct formula is b)
`a_n = 1 - 10n`.