The first term of a geometric sequencf is 3 and the sum of the second and the third term is 60.determine the first three terms and the general term Tn
T2+T3 = ar+ar^2 = 3r(r+1) = 60
r=4
...
R is 20 or r is _1
Let's solve the problem step-by-step:
Step 1: Finding the common ratio (r)
To find the common ratio (r) in a geometric sequence, we can divide any term by the previous term. In this case, we can divide the second term (let's call it a₂) by the first term (a₁).
a₂/a₁ = 60/3 (since the sum of the second and third term is 60)
a₂/a₁ = 20
So, our common ratio (r) is 20.
Step 2: Finding the second term (a₂)
We know that the first term is 3, and the common ratio is 20. Using the formula for the nth term of a geometric sequence:
a₂ = a₁ * r
a₂ = 3 * 20
a₂ = 60
Therefore, the second term (a₂) is 60.
Step 3: Finding the third term (a₃)
We know that the sum of the second and third term is 60. We already found the second term (a₂) as 60. Let's assume the third term is a₃.
a₂ + a₃ = 60
60 + a₃ = 60
a₃ = 0
Therefore, the third term (a₃) is 0.
Step 4: Determining the first three terms
Based on our calculations:
First term (a₁) = 3
Second term (a₂) = 60
Third term (a₃) = 0
So, the first three terms are 3, 60, and 0.
Step 5: Finding the general term (Tₙ)
The general term (Tₙ) of a geometric sequence can be found using the formula:
Tₙ = a₁ * r^(n-1)
In this case, a₁ = 3 (first term), r = 20 (common ratio), and n is the term number.
Therefore, the general term (Tₙ) is Tₙ = 3 * 20^(n-1).
Note: The value of n will determine the specific term you are looking for. If you provide a specific value for n, I can provide the corresponding term.
To find the first three terms of a geometric sequence, we need to determine the common ratio (r) of the sequence.
Let's assume the second term of the sequence is a, and the third term is ar.
Given that the first term is 3, we have:
a₁ = 3
We also know that the sum of the second and third terms is 60, so:
a + ar = 60
To solve this equation, we can rearrange it and factor out an 'a' term:
a(1 + r) = 60
Now, let's solve for 'a' by dividing both sides of the equation by (1 + r):
a = 60 / (1 + r)
We are trying to find the first three terms of the geometric sequence, so the terms are:
a₁ = 3 (given)
a₂ = a = 60 / (1 + r)
a₃ = ar = 60 / (1 + r) * r
To find the general term Tn, we need to find the nth term of the sequence.
The general term can be represented as:
Tn = a₁ * r^(n-1)
Substituting a₁ = 3, we have:
Tn = 3 * r^(n-1)
Therefore, the first three terms of the sequence are 3, 60 / (1 + r), and 60 / (1 + r) * r, and the general term Tn = 3 * r^(n-1).