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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

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

Notes

    1. In a geometric sequence, the ratio between consecutive terms is always the same
    2. Since each pair of terms has the ratio in common, we call it the common ratio
    1. 2 and 8 are the first two consecutive terms
    2. 'Consecutive' in this case means the terms come one right after the other
    3. 8 and 32 are the next two consecutive terms
    4. Make sure that you always divide the second number by the first number, otherwise you won't get the same ratio!
    5. 32 and 128 are also consecutive terms
    6. We got a ratio of 4 each time!
    1. The ratio between each set of consecutive numbers is 4
    2. So 4 is our common ratio!
    3. 'r' is a variable that represents the common ratio
    1. Making a table can make it easier to spot a pattern
    1. 'n' is a variable that represents the specific term number
    2. 'A(n)' is the value of the nth term in the sequence
    3. To get the second term, 8, we multiply the first term, 2, by the common ratio, 4
    4. We rewrote 8 as 2•4, so we can multiply this product by 4 to get the third term, 32
    5. 32 is just 2•42, so we can multiply that by 4 to get the fourth term, 128
    6. 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
    1. 'n' is a variable that represents the specific term number
    1. In our example, we multiplied the first term, 2, by 4 over and over again to get each new term
    2. To generalize, we'll call the first term in the sequence 'a'
    3. The number we kept multiplying by was the common ratio, 4
    4. Remember, before we called this 'r'
    5. And we know we need to multiply 'r' n-1 times, so n-1 will be our exponent
    1. Let's try using this formula to find the 9th term in the sequence
    1. 'n' is a variable that represents the specific term number
    2. Since we're trying to find the 9th term, 'n' will be 9
    3. 'a', our first term, is still 2
    4. 'r', our common ratio, is still 4
    5. Since we need to raise 'r' to the n-1 power, we will raise 4 to the 8th power
    6. 48 is 65,536
    1. Plugging the values into our formula, we get 131,072