Advertisements
Advertisements
Question
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
Advertisements
Solution
Since a, b, c, d are in G.P.
Again A.M. > G.M. for the first three terms
`(a + c)/2 > b` .....`("Since" sqrt(ac) = b)`
⇒ a + c > 2b ....(3)
Similarly, for the last three terms
`(b + d)/2 > c` .....`("Since" sqrt(bd) = c)`
⇒ b + d > 2c ....(4)
Adding (3) and (4), we get
(a + c) + (b + d) > 2b + 2c
a + d > b + c
APPEARS IN
RELATED QUESTIONS
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
Find :
the 12th term of the G.P.
\[\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .\]
Which term of the G.P. :
\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?\]
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the sum of the following series to infinity:
`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
Insert 6 geometric means between 27 and \[\frac{1}{81}\] .
Find the geometric means of the following pairs of number:
2 and 8
Find the geometric means of the following pairs of number:
−8 and −2
If the fifth term of a G.P. is 2, then write the product of its 9 terms.
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
For the G.P. if r = `1/3`, a = 9 find t7
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
The numbers 3, x, and x + 6 form are in G.P. Find nth term
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 10 years.
For a G.P. a = 2, r = `-2/3`, find S6
For a G.P. If t4 = 16, t9 = 512, find S10
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
Find : `sum_("n" = 1)^oo 0.4^"n"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
