Question 1: If
Answer:
on both sides we get
Answer:
on both sides we get
Question 3: Find such that
form three consecutive terms of a G.P.
Answer:
form three consecutive terms of a G.P.
Question 4: Three numbers are in A.P. and their sum is . If
be added to them respectively, they form a G.P. Find the numbers.
Answer:
Let the first term of the A.P. be and the common difference be
… … … … … i)
are in G.P
Substituting from i) we get
. Then A.P. is
. Then A.P. is
Question 5: The sum of three numbers which are consecutive terms of an A.P. is . If the second number is reduced by
and the third is increased by
we obtain three consecutive terms of a G.P. Find the numbers.
Answer:
in A.P.
… … … … … i)
Substituting we get
Question 6: The sum of three numbers in A.P. is
. If
are each increased by
is increased by
, the new numbers form a G.P. Find
.
Answer:
in A.P.
… … … … … i)
Substituting we get
Question 7: The sum of three numbers in G.P. is . If we subtract
from these numbers in that order, we obtain an A.P. Find the numbers.
Answer:
Let the three number be
… … … … … i)
… … … … … ii)
From i) and ii) we get
Question 8: If prove that:
i)
ii)
iii)
iv)
v)
Answer:
i)
Therefore
To prove:
ii)
Therefore
To prove:
iii)
Therefore
To prove:
iv)
Therefore
To prove:
v)
Therefore
To prove:
Question 9: If prove that:
i)
ii)
iii)
Answer:
i) Therefore
… … … … … i)
… … … … … ii)
To prove:
ii) Therefore
… … … … … i)
… … … … … ii)
To prove:
iii) Therefore
… … … … … i)
… … … … … ii)
To prove:
Question 10: If prove that the following are also in G.P.:
i)
ii)
iii)
Answer:
i)
Therefore
Therefore
ii)
Therefore
To prove:
iii)
Therefore
Therefore
Question 11: If prove that:
i)
ii)
iii)
iv)
Answer:
i) Therefore
… … … … … i)
… … … … … ii)
To prove:
Therefore
ii) Therefore
… … … … … i)
… … … … … ii)
To prove:
iii) Let the common ratio be
For
. Hence proved.
iv) Therefore
… … … … … i)
… … … … … ii)
To prove:
Question 12: If then prove that
Answer:
… … … … … i)
RHS . Hence proved.
Question 13: If then prove that
Answer:
… … … … … i)
RHS
Hence proved.
Question 14: If the ,
terms of a G.P. are
respectively. Prove that
Answer:
,
terms of a G.P. are
respectively
are in G .P.
Question 15: If then prove that
Answer:
… … … … … i)
… … … … … ii)
To prove:
Question 16: If terms of an A.P. be in G.P., then prove that
Answer:
terms of an A.P. be in G.P. Therefore
… … … … … i)
… … … … … ii)
… … … … … iii)
… … … … … iv)
Subtracting ii) from i) we get
… … … … … v)
Subtracting iii) from ii) we get
… … … … … vi)
Subtracting iv from iii) we get
… … … … … vii)
From vi) we get
Question 17: If are three consecutive terms of an A.P., prove that
are the three consecutive terms of a G.P.
Answer:
are three consecutive terms of an A.P. Therefore
are the three consecutive terms of a G.P.
Question 18: If , then prove that
Answer:
Take of both sides
… … … … … i)
… … … … … ii)
From i) and ii) we get
Question 19: If
, prove that
Answer:
… … … … … i)
… … … … … ii)
Question 20: If and,
show that
Answer:
… … … … … i)
… … … … … ii)
… … … … … iii)
Question 21: If show that
Answer:
To prove:
Question 22: If are three distinct real numbers in G.P. and.
, then prove that either
, or
.
Answer:
Let be the common ratio.
,
,
We know is always real
Case 1:
Case 2:
But as
are distinct numbers.
Therefore
Question 23: If terms of an A.P. and G.P. are both
respectively, show that
Answer:
terms of an A.P. and G.P. are both
respectively
Let be the first term and
be the common difference, then
… … … … … i)
… … … … … ii)
… … … … … iii)
Let be the first term and
be the common ratio
… … … … … iv)
… … … … …v)
… … … … … vi)
Subtracting ii) from i) we get … … … … … vii)
Subtracting iii) from ii) we get … … … … … viii)
Subtracting i) from iii) we get … … … … … xi)
To prove:
RHS