Question

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Write a recursive formula for an, the nth term of the sequence 1, -2, -5,…

Answer 1

We need two things to write the recursive formula

• The first term,
  `a`

• The common difference,
  `d`

`a` is the first value of this sequence. For this set of values,
`a = 1`, since
`1` appears first.

`d`, the common difference, is
`d = t_(3) - t_(2) = t_(2) - t_(1)`. Basically it's just the the higher term minus the previous terms. To solve for
`d`,

• 
  `d = t_(3) - t_(2) = -5 - (-2) = -5 + 2 = -3`
• 
  `d = t_(2) - t_(1) = -2 - 1 = -3`

So regardless of what terms you choose, the common difference will be the same. Now the general formula for a recursive function is

• 
  `t_(n) = a + (n - 1)d`

where
`n` is the
`nth` term. Let's substitute for
`a` and
`d` in this formula.

`t_(n) = 1 + (n - 1) * -3\\t_(n) = 1 + -3(n - 1)\\t_(n) = 1 - 3(n - 1)`

So the recursive formula is
`t_(n) = 1 - 3(n -1)`