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How do i find the nth term for the pattern; 3, 9, 27, 81, 243

What is the "nth term" formula for the pattern and how do you find it:

3, 9, 27, 81, 243

( i understand that the number is multiplied by 3 each time, but i don't know how to find the nth term :/)

AND for: 1, 3, 15, 61, 213

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1 Answer

The "nth" term can be anything, depending on n's value.  So we need to write an equation valid for any value of n.

Let's see... we know the 1st term, 3, and the 2nd is 9.  You've noted that each successive term is 3 times greater than the one before.  Since we start with 3, every term can be written as a number of 3s:

1st term: 3 = 3

2nd term: 9 = 3 * 3

3rd term: 27 = 3 * 3 * 3

4th term: 81 = 3 * 3 * 3 * 3

You'll note that for each term, the number of threes multiplied together equals the ordinal position of the term.  So for n=4, we need to multiply four threes together.  This can be written using a base and exponent to represent the number of threes we are multiplying.

For each term n, the answer is 3n, or three raised to the n-th power.  The n-th term is determined by multiplying n threes together.

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