Advertisements
Advertisements
Question
If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
Options
(a) AP
(b) GP
(c) HP
(d) none of these
Advertisements
Solution
(a) AP
\[\text{ a, b and c are in G . P } . \]
\[ \therefore b^2 = ac\]
\[\text{ Taking log on both the sides }: \]
\[2\log b = \log a + \log c . . . . . . . . \left( i \right)\]
\[Now, a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z} \]
\[\text{ Taking \log on both the sides }: \]
\[\frac{\log a}{x} = \frac{\log b}{y} = \frac{\log c}{z} . . . . . . . . \left( ii \right)\]
\[\text{ Now, comparing } \left( i \right) \text{ and } \left( ii \right): \]
\[\frac{\log a}{x} = \frac{\log a + \log c}{2y} = \frac{\log c}{z}\]
\[ \Rightarrow \frac{\log a}{x} = \frac{\log a + \log c}{2y} \text{ and } \frac{\log a}{x} = \frac{\log c}{z}\]
\[ \Rightarrow \log a \left( 2y - x \right) = xlog c \text{ and } \frac{\log a}{\log c} = \frac{x}{z}\]
\[ \Rightarrow \frac{\log a}{\log c} = \frac{x}{\left( 2y - x \right)} \text{ and } \frac{\log a}{\log c} = \frac{x}{z}\]
\[ \Rightarrow \frac{x}{\left( 2y - x \right)} = \frac{x}{z}\]
\[ \Rightarrow 2y = x + z\]
\[\text{ Thus, x, y and z are in A . P } . \]
\[\]
APPEARS IN
RELATED QUESTIONS
The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1 = a2, an = an − 1 + an − 2, n > 2
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Which term of the A.P. 84, 80, 76, ... is 0?
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.
\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]
\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If a, b, c is in A.P., prove that:
a3 + c3 + 6abc = 8b3.
A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?
A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
A man starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?
A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
Write the common difference of an A.P. the sum of whose first n terms is
Write the sum of first n odd natural numbers.
Write the sum of first n even natural numbers.
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p + q) terms is `(p + q)/2[a + b + (a - b)/(p - q)]`.
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
If n AM's are inserted between 1 and 31 and ratio of 7th and (n – 1)th A.M. is 5:9, then n equals ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
