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What is the Explicit Formula for the nth Term in a Geometric Sequence?
What is the formula for the nth term in a geometric sequence?
Summary
- 'r' represents the common ratio
- The term number is the position of the term in the sequence
- 'A(n)' gives the value of the nth term in the sequence
- 'a' represents the first term in the sequence
- This formula can be used to quickly find a specific term in a geometric sequence
- The nth term can be found by multiplying the first term, 'a', by the common ratio, 'r', n-1 times
- 'n-1' is one less than the number of the term you're trying to find
- To find the 9th term of the sequence, 'n' will be 9
- Since we're using the same sequence as before, 'a' is still 2 and 'r' is still 4

Notes
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- In a geometric sequence, the ratio between consecutive terms is always the same
- Since each pair of terms has the ratio in common, we call it the common ratio
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- 2 and 8 are the first two consecutive terms
- 'Consecutive' in this case means the terms come one right after the other
- 8 and 32 are the next two consecutive terms
- Make sure that you always divide the second number by the first number, otherwise you won't get the same ratio!
- 32 and 128 are also consecutive terms
- We got a ratio of 4 each time!
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- The ratio between each set of consecutive numbers is 4
- So 4 is our common ratio!
- 'r' is a variable that represents the common ratio
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- Making a table can make it easier to spot a pattern
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- 'n' is a variable that represents the specific term number
- 'A(n)' is the value of the nth term in the sequence
- To get the second term, 8, we multiply the first term, 2, by the common ratio, 4
- We rewrote 8 as 2•4, so we can multiply this product by 4 to get the third term, 32
- 32 is just 2•42, so we can multiply that by 4 to get the fourth term, 128
- To find the nth term in the sequence, we need to multiply the first term by 4 n-1 times, or one less than the value of the term number
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- 'n' is a variable that represents the specific term number
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- In our example, we multiplied the first term, 2, by 4 over and over again to get each new term
- To generalize, we'll call the first term in the sequence 'a'
- The number we kept multiplying by was the common ratio, 4
- Remember, before we called this 'r'
- And we know we need to multiply 'r' n-1 times, so n-1 will be our exponent
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- Let's try using this formula to find the 9th term in the sequence
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- 'n' is a variable that represents the specific term number
- Since we're trying to find the 9th term, 'n' will be 9
- 'a', our first term, is still 2
- 'r', our common ratio, is still 4
- Since we need to raise 'r' to the n-1 power, we will raise 4 to the 8th power
- 48 is 65,536
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- Plugging the values into our formula, we get 131,072