Find the nth term rule for 34,25,16,7,−2,...

Question

Rachael Singh · Accepted Answer

First of all, we need to see that the rule of the sequence is "subtract nine", and I suppose that this is the easy part.
Now, we need to see the sequence with the point of view of the first term: we start with 34, and obtain 25 by subtracting 9. Now, what's the link between the first and the third element? We obtain 16 by subtracting 9 again from 25, which means that we are subtracting 9 twice from 34.
This should lead you to the general rule: start with a number, subtract 9 once to obtain the second number, subtract 9 twice to obtain the third number...and so on.
Using a proper notation, if we define

a_{0}

as the first term of the sequence, we can write

a_{1} = a_{0} - 9

a_{2} = a_{1} - 9 = (a_{0} - 9) - 9 = a_{0} - 2 \cdot 9

a_{3} = a_{2} - 9 = (a_{0} - 2 \cdot 9) - 9 = a_{0} - 3 \cdot 9

⋮

a_{n} = a_{0} - n \cdot 9

which would be your closed formula for

a_{n}

NB: if you choose to start with

a_{1}

instead of

a_{0}

, then you must rescale your coefficients, and you would have

a_{n} = a_{1} - (n - 1) \cdot 9

Find the nth term rule for 34,25,16,7,−2,...

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