in a geometric sequence the first term is 48 and the fifth term is 3-Find the first 5 terms
a = 48
term(5) = ar^4
3 = 48r^4
r^4 = 1/16
r = ± 1/2
sequence is
48 24 12 6 3 ...
or
48 - 24 12 -6 3 ...
To find the first 5 terms of a geometric sequence, we use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1),
where:
an = nth term,
a1 = first term,
r = common ratio,
n = number of terms.
Given that the first term (a1) is 48 and the fifth term is 3, we need to find the common ratio (r) first. We can use the formula for the fifth term to do this.
3 = 48 * r^(5-1),
Simplifying the equation, we have:
3 = 48 * r^4.
Next, divide both sides of the equation by 48:
3/48 = r^4.
Reducing the fraction, we have:
1/16 = r^4.
To find the value of r, take the fourth root of both sides:
r = ∛∛(1/16) = 1/2.
Now that we have the common ratio (r = 1/2), we can find the first 5 terms by substituting the values into the formula:
a1 = 48, r = 1/2, and n = 1, 2, 3, 4, 5.
a1 = 48 * (1/2)^(1-1) = 48 * (1/2)^0 = 48 * 1 = 48.
a2 = 48 * (1/2)^(2-1) = 48 * (1/2)^1 = 48 * (1/2) = 24.
a3 = 48 * (1/2)^(3-1) = 48 * (1/2)^2 = 48 * (1/4) = 12.
a4 = 48 * (1/2)^(4-1) = 48 * (1/2)^3 = 48 * (1/8) = 6.
a5 = 48 * (1/2)^(5-1) = 48 * (1/2)^4 = 48 * (1/16) = 3.
Therefore, the first 5 terms of the geometric sequence are: 48, 24, 12, 6, and 3.
To find the first 5 terms of a geometric sequence, we need to identify the common ratio of the sequence. The common ratio is found by dividing any term by its preceding term. In this case, we can use the second term (since we know the first term) and divide it by the first term.
Let's assume the second term is denoted as "a2". The common ratio (r) is then given by:
r = a2 / a1
Given that the first term (a1) is 48, we can calculate the common ratio as follows:
r = (3 - a1) / a1
Now we substitute a1 = 48 into the equation:
r = (3 - 48) / 48
r = -45 / 48
r = -15 / 16
So, the common ratio (r) is -15/16.
Now that we have the common ratio, we can find the first 5 terms. The formula for the nth term of a geometric sequence is:
an = a1 * r^(n-1)
Using this formula, we can calculate the first five terms (a1, a2, a3, a4, a5):
a1 = 48
a2 = a1 * r
a3 = a2 * r
a4 = a3 * r
a5 = a4 * r
Substituting the values:
a1 = 48
a2 = 48 * (-15/16)
a3 = (48 * (-15/16)) * (-15/16)
a4 = ((48 * (-15/16)) * (-15/16)) * (-15/16)
a5 = (((48 * (-15/16)) * (-15/16)) * (-15/16)) * (-15/16)
Now we can calculate the values of the first five terms.