An arithmetic and a geometric sequence have the same first terms.(2)....and the same second term say X..The sum of the first 3 terms of the arithmetic sequence equals to the third term of the geometric sequence.Calculate the first 3 terms of each sequences
a+d = ar
3/2 (2a+2d) = ar^2
3a+3d = ar^2
3(a+d) = ar^2
3ar = ar^2
3r = r^2
r = 3
So,
a+d = 3a
d = 2a
The sequences are thus
a,3a,5a,7a,9a,...
a,3a,9a,27a,...
Note that a+3a+5a = 9a, as required
Pick a value for a, and plug and chug.
Let's denote the common first term of both sequences as a.
Arithmetic sequence: a, a + d, a + 2d, ...
Geometric sequence: a, a * r, a * r^2, ...
Given that the second term of both sequences is X, we can equate them:
a + d = X ---> (1)
a * r = X ---> (2)
Now, let's consider the sum of the first 3 terms of the arithmetic sequence:
(a) + (a + d) + (a + 2d) = a + (a + d) + (a + 2d) = 3a + 3d
We are told that this sum is equal to the third term of the geometric sequence, which is a * r^2:
3a + 3d = a * r^2 ---> (3)
To solve for the values of a, d, r, and X, we need to solve the system of equations (1), (2), and (3).
From equation (1), we have: a + d = X ---> d = X - a
Substituting this value of d in equation (3), we get:
3a + 3(X - a) = a * r^2
Expanding:
3a + 3X - 3a = a * r^2
Cancelling out the common terms:
3X = a * r^2 ---> (4)
Now, we have equations (2) and (4) in terms of a, r, and X. We can solve them simultaneously.
From equation (2), we have: a * r = X
Substituting this value of X in equation (4), we get:
3(a * r) = a * r^2
Dividing both sides by (a * r):
3 = r
Now we know that r = 3.
Substituting this value of r in equation (2), we get:
a * 3 = X
Dividing both sides by 3:
a = X/3
Now that we know the values of a and r, we can find the first three terms of both sequences.
Arithmetic sequence: a, a + d, a + 2d
Using d = X - a, we have:
a, a + (X - a), a + 2(X - a)
Simplifying each term:
a, X, 2X - a
Geometric sequence: a, a * r, a * r^2
Using r = 3, we have:
a, a * 3, a * 3^2
Simplifying each term:
a, 3a, 9a
So, the first three terms of each sequence are:
Arithmetic sequence: a, X, 2X - a
Geometric sequence: a, 3a, 9a
To find the first three terms of each sequence, let's first denote the common ratio of the geometric sequence as 'r' and the common difference of the arithmetic sequence as 'd'.
Given:
First term of both sequences = 2
Second term of both sequences = X
The arithmetic sequence can be represented as:
Term 1: 2
Term 2: 2 + d = X (1)
Term 3: 2 + 2d (2)
The geometric sequence can be represented as:
Term 1: 2
Term 2: 2 * r = X (3)
Term 3: X * r (4)
To find the values of 'd' and 'r', we can use the given information.
From equation (1):
2 + d = X
d = X - 2
From equation (3):
2 * r = X
r = X / 2
Substituting the value of 'r' into equation (4):
Term 3 of geometric sequence = X * (X / 2) = X^2 / 2 (5)
Given that the sum of the first three terms of the arithmetic sequence is equal to the third term of the geometric sequence, we can equate equations (2) and (5):
2 + 2d = X^2 / 2
Substituting the value of 'd' from equation (1):
2 + 2(X - 2) = X^2 / 2
Simplifying the equation:
4 + 2X = X^2 / 2
Multiplying both sides by 2:
8 + 4X = X^2
Rearranging the equation:
X^2 - 4X - 8 = 0
Using the quadratic formula, we can solve for 'X':
X = (-(-4) ± sqrt((-4)^2 - 4 * 1 * -8)) / (2 * 1)
X = (4 ± sqrt(16 + 32)) / 2
X = (4 ± sqrt(48)) / 2
X = (4 ± 4√3) / 2
X = 2 ± 2√3
Now that we have the values of 'X', we can substitute them back into the equations to find the first three terms of each sequence.
For X = 2 + 2√3:
Term 1 of arithmetic sequence = 2
Term 2 of arithmetic sequence = X = 2 + 2√3
Term 3 of arithmetic sequence = 2 + 2(2 + 2√3) = 6 + 4√3
Term 1 of geometric sequence = 2
Term 2 of geometric sequence = X = 2 + 2√3
Term 3 of geometric sequence = X * r = (2 + 2√3)(2√3) = 4√3 + 12
Therefore, when X = 2 + 2√3, the first three terms of each sequence are as follows:
Arithmetic sequence: 2, 2 + 2√3, 6 + 4√3
Geometric sequence: 2, 2 + 2√3, 4√3 + 12
Similarly, you can repeat the above steps for X = 2 - 2√3, which will give another set of values for the first three terms of each sequence.