The second and sixth terms of a G.P are 7
and 243/18, what is their common ratio?
A. 2.48
B. 3.68
C. 4.98
D. 1.18
Let's assume that the common ratio of the GP is "r".
The second term of the GP is given as 7, so we can write the second term as:
7 = a * r^(2-1)
7 = a * r
Similarly, the sixth term of the GP is given as 243/18, so we can write the sixth term as:
243/18 = a * r^(6-1)
243/18 = a * r^5
Dividing the equations, we get:
(243/18)/(7) = (a * r^5) / (a * r)
243/126 = r^4
Taking the fourth root on both sides, we get:
(r^4)^(1/4) = (243/126)^(1/4)
r = (243/126)^(1/4)
r ≈ 1.18
Therefore, the common ratio is approximately 1.18, so the answer is D. 1.18.
To find the common ratio (r) of a geometric progression (G.P.), we can use the formula:
\[r = \sqrt[n-1]{\frac{a_n}{a_1}}\]
where \(a_n\) is the nth term and \(a_1\) is the first term of the G.P.
In this case, the second term is 7 and the sixth term is \(\frac{243}{18}\).
Let's substitute these values into the formula:
\[r = \sqrt[6-1]{\frac{\frac{243}{18}}{7}}\]
Simplifying the fraction:
\[r = \sqrt[5]{\frac{243}{18} \times \frac{1}{7}}\]
\[r = \sqrt[5]{\frac{243}{126}}\]
To simplify the fraction further, we can divide both the numerator and denominator by 9:
\[r = \sqrt[5]{\frac{27}{14}}\]
Calculating the 5th root of the fraction:
\[r \approx 1.180\]
Therefore, the common ratio is approximately 1.180.
Hence, the correct option is D. 1.18.