Question 1: Show that each one of the following progressions is a G.P.. Also, find the common ratio in each case:
i)
ii)
iii)
iv)
Answer:
i)
Given
and
and
Therefore and
are in G.P. where
and
ii)
Given
Therefore are in G.P. where
and
iii)
Given
and
Therefore are in G.P. where
is the first term and
iv)
Given
and
and
Therefore and
are in G.P. where
and
Question 2: Show that the sequence defined by
is a G.P.
Answer:
Given
is a G.P
Given
and
and
Therefore
is a G.P where
and
Question 3: Find:
i) the term of the G.P.
ii) the term of the G.P.
iii) the term of the G.P.
iv) the term of the G.P.
v) the term of the G.P.
vi) the term of the G.P.
Answer:
i) To find the term of the G.P.
Here
We know,
ii) To find the term of the G.P.
Here
We know,
iii) To find the term of the G.P.
Here
We know,
iv) To find the term of the G.P.
Here
We know,
v) To find the term of the G.P.
Here
We know,
vi) To find the term of the G.P.
Here
We know,
Question 4: Find the term from the end of the G.P.
Answer:
Here
Last term
Now, when we reverse the G.P., we have , and
We know,
Question 5: Which term of the progression is
Answer:
Given series: is
Here
Let be the
term
Therefore is the
term of the given G.P.
Question 6: Which term of the G.P.:
i)
ii)
iii)
iv)
Answer:
i) Given series:
?
Here
We know
Let be the
term
Therefore is the
term of the given G.P.
ii) Given series:
Here
We know
Let be the
term
Therefore is the
term of the given G.P.
iii) Given series:
Here
We know
Let be the
term
Therefore is the
term of the given G.P.
iv) Given series:
Here
We know
Let be the
term
Therefore is the
term of the given G.P.
Question 7: Which term of the progression is
?
Answer:
Given series
Here
We know
Let be the
term
Therefore is the
term of the given G.P.
Question 8: Find the terms from the end of the G.P.
Answer:
Given series:
Reversing the G.P. we get
We know
Question 9: The term of the G.P. is
and the
term is
, find the G.P.
Answer:
Given … … … … … i)
Given … … … … … ii)
Dividing ii) by i) we get
Substituting in i) we get
Therefore the G.P. is
Question 10:The term of the G.P. is
times the
term and
term is
. Find the G.P.
Answer:
Given
Substituting we get
Therefore the G.P. is
Question 11: If the G.P.’s and
have their
terms equal, find the value of
.
Answer:
For G.P.
Therefore
We know
… … … … … i)
For G.P.
Therefore
We know
… … … … … ii)
From i) and ii) we get
Question 12: The and
term of a G.P. are
, and
respectively, prove tht
.
Answer:
Given … … … … … i)
… … … … … ii)
… … … … … iii)
Therefore . Hence proved.
Question 13: The term of a G.P. is square of its
term and the first term is
. Find its
term.
Answer:
Given and
Question 14: In a G.P. the term is
and the
term is
. Find the
term.
Answer:
… … … … … i)
… … … … … ii)
Dividing ii) and i) we get
Substituting in i) we get
Question 15: If and
are different real numbers such that
, then show that
and
are in G.P.
Answer:
Given
Since all the terms are square, therefore they cannot be less than zero. Hence,
are in G.P.
Question 16: If
, then show that
and
are in G.P.
Answer:
Given
Take the first two terms
Apply componendo and dividendo we get
… … … … … i)
Similarly, Take the last two terms
Apply componendo and dividendo we get
… … … … … ii)
Therefore and
are in G.P.
Question 17: If the and the
terms of a G.P. are
and
respectively, show that
term is
Answer:
Given … … … … … i)
… … … … … ii)
Dividing i) by ii)
… … … … … i)
Substituting the value of r in ii) we get
… … … … … ii)
Therefore … … … … … iii)
Substituting i) and ii) in iii) we get