In a g.p second term is 1/9 and seventh term is 1/2187 then find the series
(ar^6)/(ar) = r^5 = 9/2187 = 1/243
r = 1/3
so, a = 1/3
To find the series of a geometric progression (G.P.), we need to determine the common ratio (r) and the first term (a).
Given that the second term is 1/9, we can use the formula for the nth term of a G.P.:
an = a * r^(n-1)
Substituting n=2 and a=1/9, we have:
1/9 = a * r^(2-1)
1/9 = a * r
Next, we are given that the seventh term is 1/2187. Substituting n=7 and a=1/9 into the G.P. formula:
1/2187 = (1/9) * r^(7-1)
1/2187 = (1/9) * r^6
Now, we have two equations:
1/9 = a * r
1/2187 = (1/9) * r^6
To solve these equations simultaneously, we can divide the second equation by the first equation:
(1/2187) / (1/9) = (1/9) * r^6 / (1/9)
1/2187 * 9 = r^6
Now, let's simplify:
1/243 = r^6
To get the value of r, we can take the 6th root of both sides:
r = (1/243)^(1/6)
r = 1/3
Now that we have the common ratio (r), we can substitute it back into the first equation to find the first term (a):
1/9 = a * (1/3)
1/9 = a/3
a = 1/3 * 1/9
a = 1/27
Therefore, the series is: 1/27, 1/81, 1/243, 1/729, ...
Each term is obtained by multiplying the previous term by 1/3.