if the 3rd,5th and 11th terms are distinct and form a geometric sequence, what is the 2020th term if then 4th term is 5?
please show how to solve the problem...
a + 3d = 5 ... a = 5 - 3d
(a + 4d) / (a + 2d) = (a + 10d) / (a + 4d) ... "cross" multiply
a^2 + 8ad + 16d^2 = a^2 + 12ad + 20d^2 ... subtract a^2
8ad + 16d^2 = 12ad + 20d^2 ... divide by 2d
4a + 8d = 6a + 10d ... substituting ... 20 - 12d + 8d = 30 - 18d + 10d
solve for d , then substitute back to find a
Well, well, well, it seems we have a geometric sequence on our hands! Let me put on my clown shoes and calculate the 2020th term for you.
Since the 4th term is 5, let's call the common ratio between the terms "r."
We know that the 5th term is the 4th term (5) multiplied by "r." So, we have 5r as the 5th term.
Similarly, the 11th term is the 5th term (5r) multiplied by "r" again. Therefore, the 11th term is 5r^2.
Given that the 3rd, 5th, and 11th terms are distinct, we can set up the equation:
5r^2 = 3r
To solve for "r," we divide both sides by "r":
5r = 3
Dividing both sides by 5, we find:
r = 3/5
Now, we want to find the 2020th term. Since we know the common ratio (r = 3/5), we can find the nth term using the formula:
tn = t1 * r^(n-1)
Plugging in the values, we have:
t2020 = 5 * (3/5)^(2020-1)
Now, if you're expecting me to calculate that massive power, I'm sorry to disappoint you. My clown powers only go up to quadratic equations, not exponential ones. But hey, try popping that formula into a calculator, and you'll find the answer to the 2020th term! Good luck!
To find the 2020th term of the geometric sequence, we first need to determine the common ratio of the sequence.
Let's denote the first term as 'a' and the common ratio as 'r' (where a ≠ 0 and r ≠ 0).
We are given that the 4th term is 5, which means the sequence is: a, ar, ar^2, 5.
Using this information, we can write the equation: ar^2 = 5.
Next, we're given that the 3rd, 5th, and 11th terms are distinct and form a geometric sequence. Therefore, we can write the following equations:
For the 3rd and 5th terms: ar = ar^2/r, which simplifies to r = 1/r.
For the 5th and 11th terms: ar^2 = ar^4/r, which simplifies to r^3 = 1.
Now, let's solve these two equations:
r = 1/r
=> r^2 = 1
=> r = ±1
r^3 = 1
=> r = 1
Since r cannot be ±1 simultaneously, we conclude that r = 1.
Now, substituting r = 1 back into the equation ar^2 = 5, we have:
a = 5/1^2
=> a = 5
So, we know that the first term 'a' is 5 and the common ratio 'r' is 1.
To find the 2020th term, we can use the formula for the n-th term of a geometric sequence:
Tn = a * r^(n-1)
Plugging in the values, we have:
T2020 = 5 * 1^(2020-1)
=> T2020 = 5 * 1^2019
=> T2020 = 5 * 1
=> T2020 = 5
Therefore, the 2020th term of the geometric sequence is 5.