The first, third and fifth terms of a geometric sequence from arithmetic sequence. If the first term of the sequence is 3, find the 10th term of the geometric sequence
ar^2 - a = ar^4 - ar^2
but a = 3, so
3r^2 - 3 = 3r^4 - 3r^2
3r^4 - 6r^2 + 3 = 0
r^4 - 2r^2 + 1 = 0
(r^2 - 1)^2 = 0
r^2 = 1
r = ± 1
If r = 1, the terms of the GP would be 3, 3, 3, 3, ...
if r = -1, the terms of the GP would be 3, -3, 3, -3, ...
term 10 could be either 3, or -3
A sequence with a common ratio of 1 is technically a GP but is of
little interest, the same is true if r = -1, then the same terms would simple alternate between + and -1
Well, well, well, looks like we have a combination of arithmetic and geometric sequences here! It's like the math equivalent of a mashup song! Let's get down to business.
If the first term of the sequence is 3, we know that the third term is also part of the arithmetic sequence. So, let's determine the common difference, shall we?
Since the first term is 3 and the fifth term is part of the geometric sequence, let's label it as "a" instead of a specific value. The arithmetic sequence formula gives us:
a = 3 + (3rd term - 1) * d
As for the geometric sequence, the ratio between consecutive terms should be the same. We'll call this ratio "r".
The fifth term, a, can be expressed as:
a = 3 * r^4
To make things easier, let's find the value of "r" by dividing the fifth term by the third term:
r = (3 * r^4) / (3 * r^2)
r^3 = 1
r = 1
Well, well, well, it seems like our geometric sequence is actually pretty basic! With a common ratio of 1, all terms will be the same value.
Therefore, the 10th term will also be 3, just like the first term! There you go, 3 is the answer, my friend!
To find the 10th term of a geometric sequence, we need to first find the common ratio (r) of the sequence.
Given that the first, third, and fifth terms form an arithmetic sequence, we can use this information to find the common difference (d) of the arithmetic sequence.
Let's find the common difference (d):
The first term of the arithmetic sequence is 3.
The third term of the arithmetic sequence is the second term (a_2) plus the common difference (d). Since the geometric sequence shares the terms of the arithmetic sequence, the third term (a_3) of the geometric sequence is a_2 + d.
Similarly, the fifth term of the arithmetic sequence is a_4 + d.
We are given that a_1 = 3, a_3 = a_2 + d, and a_5 = a_4 + d.
Using this information, we can form the following equations:
a_1 = 3
a_3 = a_2 + d
a_5 = a_4 + d
Given that a_1 = 3, we can substitute this value into the other equations:
3 = a_2 + d -------------- (Equation 1)
a_3 = a_2 + d
a_4 = a_2 + 2d
a_5 = a_2 + 3d
Substituting a_2 + d for a_3 and a_2 + 2d for a_4 in Equation 1, we can simplify the equation to:
3 = (a_2 + d) + d
3 = a_2 + 2d
Substituting a_2 + 2d for a_4 in the equation a_5 = a_2 + 3d, we get:
a_5 = (a_2 + 2d) + d
a_5 = a_2 + 3d
Since we now have two equations in terms of a_2 and d, we can solve for the values of a_2 and d.
From Equation 1, we have:
3 = a_2 + 2d
Simplifying, we get:
a_2 = 3 - 2d -------------- (Equation 2)
From Equation a_5 = a_2 + 3d, we have:
a_5 = a_2 + 3d
Substituting the value of a_2 from Equation 2, we get:
a_5 = 3 - 2d + 3d
a_5 = 3 + d -------------- (Equation 3)
Now, we can set Equations 2 and 3 equal to each other and solve for d:
3 - 2d = 3 + d
Subtracting 3 from both sides, we have:
-2d = d
Adding 2d to both sides, we get:
0 = 3d
Dividing both sides by 3, we have:
0 = d
Since d = 0, we can substitute this value back into Equation 2 to solve for a_2:
a_2 = 3 - 2(0)
a_2 = 3
Therefore, the value of a_2 is 3 and the common difference (d) is 0. This means that the geometric sequence is actually just a sequence of 3's.
To find the 10th term of the geometric sequence, we can use the formula for the nth term of a geometric sequence: a_n = a_1 * r^(n-1).
Since a_1 = 3 and r = 0, the 10th term (a_10) is:
a_10 = 3 * 0^(10-1)
a_10 = 3 * 0^9
a_10 = 3 * 0
a_10 = 0
Therefore, the 10th term of the given geometric sequence is 0.
To find the 10th term of the geometric sequence, we need to first determine the common ratio (r) of the sequence.
Since the first, third, and fifth terms form an arithmetic sequence, let's find the common difference (d) between them.
The first term is 3, and the third term should be the first term plus the common difference, so let's call the third term a3. We have the equation: a3 = 3 + d.
Similarly, the fifth term a5 is the first term plus twice the common difference, so we have the equation: a5 = 3 + 2d.
Since the terms form an arithmetic sequence, we can set up the following equations:
a3 = a1 + (3-1)*d
a5 = a1 + (5-1)*d
Substituting the values we already know, we have:
3 + d = 3 + (3-1)*d = 3 + 2d
3 + 2d = 3 + (5-1)*d = 3 + 4d
From equation 1, we can simplify it to: d = 0
Now we know that the common difference (d) in the arithmetic sequence is 0, indicating that the terms of the sequence are all the same. Therefore, it is not a valid geometric sequence.
In this case, since we cannot determine the common ratio, we cannot find the 10th term of the geometric sequence.