If 3/5/9 and 40/1/2 are the first and the last terms respectively, of a geometric sequence, determine the values of the fifth term if there are seven terms in the sequence
Better clarify just what 3/5/9 means.
Hooray Ho Tim Steve
To determine the values of the fifth term of a geometric sequence, we need to find the common ratio of the sequence.
In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio. Let's start by finding the common ratio of the sequence.
Given:
- First term (a₁) = 3/5/9
- Last term (a₇) = 40/1/2
The common ratio (r) can be found by dividing any term by the previous term.
r = a₇ / a₁
Let's convert the fractions into a more manageable form before performing the calculation:
a₁ = 3/5/9 = (3/5)/(9/1) = (3/5) * (1/9) = 3/45 = 1/15
a₇ = 40/1/2 = (40/1)/(2/1) = 40/2 = 20
Now we can calculate the common ratio:
r = 20 / (1/15) = 20 * 15/1 = 300
Now that we have the common ratio, we can find the fifth term (a₅) using the formula:
a₅ = a₁ * r^(n-1)
Since there are seven terms in the sequence, n = 7.
a₅ = (1/15) * 300^(7-1)
= (1/15) * 300^6
Calculating this expression gives us the value of the fifth term in the sequence.