The second and fifth term of a geometric progression are 21 and 567 respectively. Find the first term and the common ratio of the progression

the 5th terms is the 3rd after the 2nd term, so their ratio is r^3. Thus,

r^3 = 567/21 = 27
r = 3
a = 21/3 = 7

If 8x,y-4 are in A.P find p and q

Chu

Well, since I'm a Clown Bot, how about I give you a funny response first?

Why did the geometric progression go to the circus?

Because it wanted to find its first term and common ratio under the big top!

Now, to answer your question seriously, let's consider the formula for the nth term of a geometric progression:

an = a1 * r^(n-1)

We have two pieces of information:
- The second term, a2, is 21
- The fifth term, a5, is 567

Let's use these values to set up a couple of equations:

21 = a1 * r^(2-1)
567 = a1 * r^(5-1)

Simplifying, we can rewrite these equations as:

21 = a1 * r
567 = a1 * r^4

Now, let's divide the second equation by the first equation:

567 / 21 = (a1 * r^4) / (a1 * r)

This simplifies to:

27 = r^3

Taking the cube root of both sides, we find:

r = 3

Now that we know the common ratio, we can substitute it back into either of the original equations to find the first term:

21 = a1 * 3

Solving for a1, we get:

a1 = 7

So, the first term of the progression is 7 and the common ratio is 3.

I hope I've brought a smile to your face with my clownish approach to answering your question! Let me know if there's anything else I can assist you with.

To find the first term and common ratio of a geometric progression, we can use the formula:

\(a_n = a_1 \cdot r^{(n - 1)}\)

where \(a_n\) is the \(n\)th term of the geometric progression, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term.

Given that the second term (\(a_2\)) is 21 and the fifth term (\(a_5\)) is 567, we can set up two equations using the formula above:

Equation 1:
\(a_2 = a_1 \cdot r^{(2 - 1)} = a_1 \cdot r\)

Equation 2:
\(a_5 = a_1 \cdot r^{(5 - 1)} = a_1 \cdot r^4\)

Substituting the given values into these equations, we get:

Equation 1:
\(21 = a_1 \cdot r\)

Equation 2:
\(567 = a_1 \cdot r^4\)

Now, we can solve these two equations to find the values of \(a_1\) (the first term) and \(r\) (the common ratio).

Dividing Equation 2 by Equation 1, we get:

\(\frac{{567}}{{21}} = \frac{{a_1 \cdot r^4}}{{a_1 \cdot r}} = \frac{{a_1}}{{a_1}} \cdot \frac{{r^4}}{{r}} = r^3\)

Simplifying this, we have:

\(r^3 = 27\)

Now, we can take the cube root of both sides to find the value of \(r\):

\(r = \sqrt[3]{{27}} = 3\)

Substituting this value of \(r\) back into Equation 1, we can solve for \(a_1\):

\(21 = a_1 \cdot 3\)

Dividing both sides by 3, we get:

\(a_1 = \frac{{21}}{{3}} = 7\)

Therefore, the first term (\(a_1\)) of the geometric progression is 7 and the common ratio (\(r\)) is 3.