The second and fifth term of a G.p are 3/2 and 1/1.2, respectively. what is the first term.
a1 = a2/r = (3/2) / ∛((1/1.2)/(3/2)) = 3/10 ∛225
To find the first term of a geometric progression (G.P.), we need to determine the common ratio (r) and one other term in the sequence. In this case, we are given the second term and the fifth term.
Let's denote the first term as 'a', and the common ratio as 'r'.
Given: Second term = 3/2 = a * r
Also, given: Fifth term = 1/1.2 = a * r^4
To eliminate the 'a' term, let's divide the equation representing the fifth term by the equation representing the second term:
(1/1.2) / (3/2) = (a * r^4) / (a * r)
Simplifying, we get:
(1/1.2) * (2/3) = r^3
Simplifying further:
1/1.8 = r^3
To find 'r', we will raise both sides to the power of 1/3:
(1/1.8)^(1/3) = r
Calculating the value:
r ≈ 0.873
Now, we can substitute the value of 'r' into the equation for the second term to find 'a':
3/2 = a * 0.873
Solving for 'a':
a = (3/2) / 0.873
a ≈ 2.174
Therefore, the first term of the G.P. is approximately 2.174.
To find the first term of a geometric progression (G.P.), we can use the formula:
an = a1 * r^(n-1)
Here, an represents the 'n'th term of the G.P., a1 is the first term, and r is the common ratio.
Given that the second term (a2) is 3/2 and the fifth term (a5) is 1/1.2, we can set up two equations using the formula mentioned above:
a2 = a1 * r^(2-1)
a5 = a1 * r^(5-1)
Substituting the given values into the equations:
3/2 = a1 * r
1/1.2 = a1 * r^4
Now, we can solve these two equations simultaneously to find the values of a1 and r.
Divide the second equation by the first equation to eliminate 'a1':
(1/1.2) / (3/2) = (a1 * r^4) / (a1 * r)
1/(1.2*3/2) = r^3
Simplifying further:
1/(1.8/2) = r^3
2/1.8 = r^3
20/18 = r^3
10/9 = r^3
∛(10/9) = r
Taking the cube root of both sides gives us the value of r:
r ≈ 0.9949
Now we can substitute this value back into one of the equations to find a1. Let's use the first equation:
3/2 = a1 * 0.9949
Simplify for a1:
a1 = (3/2) / 0.9949
Evaluating this expression will give us the value of a1, which is the first term of the G.P.