Calculate the missing terms of the geometric sequence…5,?,?,?,80
To find the missing terms of the geometric sequence, we need to determine the common ratio (r) between the terms.
5 * r = ? (first missing term)
? * r = ? (second missing term)
? * r = ? (third missing term)
To find the common ratio, we can divide any term by its previous term. Let's use the 4th term (80) and the 3rd term (third missing term).
80 ÷ ? = ? (third missing term ÷ 80)
? = 80 ÷ (third missing term ÷ 80) (equation 1)
Now, let's use the 3rd term (third missing term) and the second term (second missing term).
? ÷ ? = ? (second missing term ÷ third missing term)
? = (second missing term ÷ third missing term) (equation 2)
Substituting equation 2 into equation 1:
80 ÷ ((second missing term ÷ third missing term) ÷ 80)
Simplifying further:
80 ÷ (80 ÷ ((second missing term ÷ third missing term))
We can simplify this expression by multiplying by the reciprocal of the fraction in the denominator:
80 * ((second missing term ÷ third missing term) ÷ 80)
The 80's cancel out:
(second missing term ÷ third missing term)
Now we have:
(second missing term ÷ third missing term) = ? (common ratio)
We can assign a variable to the common ratio, let's use x:
x = (second missing term ÷ third missing term)
Now we can find the missing terms. Let's assign the label a to the first term (5) and use the equation for the nth term of a geometric sequence:
nth term = a * r^(n-1)
We can use this equation for the 2nd, 3rd, and 4th missing terms:
2nd missing term = 5 * (x)^(2-1)
3rd missing term = 5 * (x)^(3-1)
4th missing term = 5 * (x)^(4-1)
Finally, we can calculate the missing terms using the common ratio:
2nd missing term = 5 * (x)^(1)
3rd missing term = 5 * (x)^(2)
4th missing term = 5 * (x)^(3)
However, without additional information or constraints, there are infinitely many possibilities for the missing terms, as there is no unique solution for the common ratio or the missing terms.