If 5th and 8th terms of a g.p. Be 48 and 348 respectively.find the g.p.
The second and fifth terms of a GP are _6 and 48 ,respectively. Find the sum of the first four terms
To find the geometric progression (G.P.), we need to determine the common ratio, 'r'.
Given that the 5th term of the G.P. is 48 and the 8th term is 348, we can use the formula for the nth term of a G.P. to set up two equations and solve for 'r'.
The formula for the nth term of a G.P. is:
an = a1 * r^(n-1),
where 'an' is the nth term, 'a1' is the first term, 'r' is the common ratio, and 'n' is the position of the term.
We use the given information to set up the following equations:
48 = a1 * r^(5-1),
348 = a1 * r^(8-1).
Simplifying each equation, we get:
Equation 1: 48 = a1 * r^4,
Equation 2: 348 = a1 * r^7.
Dividing Equation 2 by Equation 1, we eliminate 'a1' and get:
348/48 = (a1 * r^7) / (a1 * r^4).
Simplifying further, we have:
7.25 = r^3.
Now, taking the cube root of both sides, we find:
r = ∛(7.25).
Using a calculator, we find that the cube root of 7.25 is approximately 1.969.
So, the common ratio of the G.P. is approximately 1.969.
Now, to find the first term 'a1', we can use either Equation 1 or Equation 2.
Using Equation 1, which is simpler, we substitute the known values:
48 = a1 * (1.969)^4.
Solving this equation, we get:
a1 = 48 / (1.969)^4.
Using a calculator, we find that a1 is approximately 4.104.
Therefore, the G.P. is given by a1 = 4.104, r = 1.969.
Hence, the G.P. is:
4.104, 8.099 (approximately), 15.943 (approximately), 31.407 (approximately), ...
ar^7 = 348
ar^4 = 48
divide: r^3 = 29/4
now you can find a.