Advertisements
Advertisements
प्रश्न
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
Advertisements
उत्तर
\[\text { a, b and c are in A . P } . \]
\[ \therefore 2b = a + c . . . . . . . (i)\]
\[\text { Also, a, b and d are in G . P } . \]
\[ \therefore b^2 = ad . . . . . . . (ii)\]
\[\text { Now, } \left( a - b \right)^2 \]
\[ = a^2 - 2ab + b^2 \]
\[ = a^2 - a\left( a + c \right) + ad \left[ \text { Using (i) and (ii) } \right]\]
\[ = ad - ac\]
\[ = a\left( d - c \right)\]
\[ \Rightarrow \left( a - b \right)^2 = a\left( d - c \right)\]
\[\text { Therefore, } a, \left( a - b \right) \text { and } \left( d - c \right) \text { are in G . P }.\]
APPEARS IN
संबंधित प्रश्न
If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
Find the rational number whose decimal expansion is `0.4bar23`.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
Check whether the following sequence is G.P. If so, write tn.
3, 4, 5, 6, …
For the G.P. if r = − 3 and t6 = 1701, find a.
Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers 3, x, and x + 6 form are in G.P. Find nth term
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
For the following G.P.s, find Sn.
`sqrt(5)`, −5, `5sqrt(5)`, −25, ...
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,...`
Express the following recurring decimal as a rational number:
`0.bar(7)`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term is -
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.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
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 ______.
