The second term of a g.p. Is 27 and 7th term is 1/9. Find the first term and the common ratio.

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The second term of g.p.is 27and 7th term 1/9 find the first term and the common ratio

To find the first term and the common ratio of a geometric progression (g.p.), we need to use the given information about the second term and the seventh term.

Let's denote the first term of the g.p. as 'a' and the common ratio as 'r'.

Given, the second term is 27. We can use the formula for the n-th term of a g.p., which is:

Tn = a * r^(n-1)

Plugging in the values, we have:

27 = a * r^(2-1)
27 = a * r

Next, we are given that the seventh term is 1/9. Plugging this into the formula, we have:

(1/9) = a * r^(7-1)
(1/9) = a * r^6

Now we have a system of two equations:

27 = a * r
(1/9) = a * r^6

We can solve this system of equations to find the values of 'a' and 'r'.

Dividing the first equation by the second equation will eliminate 'a':

(27) / (1/9) = (a * r) / (a * r^6)
243 = r^(-5)
1/243 = r^5

Now, let's take the fifth root of both sides to solve for 'r':

r = (1/243)^(1/5)
r = 1/3

We have now found the value of the common ratio, which is r = 1/3.

Substituting the value of 'r' into the first equation, we can solve for 'a':

27 = a * (1/3)
81 = a

So, the first term of the g.p. is a = 81 and the common ratio is r = 1/3.

again?

ar^6 = 1/9
ar = 27

divide, and you have

r^5 = 1/3^5
r = 1/3
So, a=81