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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.

$t_{n} = 1 + (n - 1) \times -3\\t_{n} = 1 + -3(n - 1)\\t_{n} = 1 - 3(n - 1)$

So the recursive formula is $t_{n} = 1 - 3(n -1)$