Notes:
- We know
when
and
when
Question 1: Find the sum of the following geometric progressions:
i) to
terms
ii) to
terms
iii)
to
terms
iv)
to
terms
v)
to
terms
Answer:
i) Given series:
Here
ii) Given series:
Here
iii) Given series:
Here
iv) Given series:
Here
v) Given series:
Here
Question 2: Find the sum of the following geometric progressions:
i) to
terms
ii)
to
terms
iii)
to
terms
iv) to
terms
v)
to
terms
vi)
vii) to
terms
viii) to
terms
ix) to
terms
Answer:
i) Given sequence:
Here
ii) Given sequence:
Here
iii) Given sequence:
Here
iv) Given sequence:
v) Given sequence:
vi) Given sequence:
Here
vii) Given sequence:
Here
viii) Given sequence:
Here
ix) Given sequence:
Here
Question 3: Evaluate the following:
i) ii)
iii)
Answer:
i)
ii)
iii)
Question 4: Find the sum of the following series:
i) to
terms
ii) to
terms
iii) to
terms
iv) to
terms
v) to
terms
Answer:
i) to
terms
ii) to
terms
iii) to
terms
iv) to
terms
v) to
terms
Question 5: How many terms of the G.P. to be taken together to make
?
Answer:
Here we have
Question 6: How many terms of the series must be taken to make the sum equal to
?
Answer:
Here we have
Question 7: How many terms of the sequence must be taken to make the sum
?
Answer:
Here we have
Question 8: The sum of terms of the G.P. is
is 381. Find the value of
.
Answer:
Here we have
Question 9: The common ratio of a G.P. is and the last term is
. If the sum of these terms be
, find the first term.
Answer:
Here we have
We know
… … … … … i)
Also,
… … … … … ii)
Dividing ii) by i) we get
Question 10: The ratio of the sum of first three terms is to that of first six terms of a G.P. is . Find the common ratio.
Answer:
Given
Let the first term be and the common ratio is
Let
or
. But
Hence
Question 11: The and the
term of a G.P. are
and
respectively. Find the sum of n terms of the G.P.
Answer:
Given
and
… … … … … i)
and
… … … … … ii)
Dividing ii) by i) we get
Substituting in i) we get
Question 12: Find the sum
Answer:
Question 13: The fifth term of a G.P. is whereas its second term is
. Find the series and sum of its first eight terms.
Answer:
Let the first term be and the common ratio be
Given
… … … … … i)
Similarly,
… … … … … ii)
Dividing ii) by i) we get
Substituting in i) we get
Question 14: If be respectively the sums of
terms of a G.P., then prove that
.
Answer:
Let the first term be and the common ratio be
Given
To prove:
LHS
RHS. Hence proved.
Question 15: Show that the ratio of the sum of the first n terms of a G.P. to the sum of terms from to
term is
.
Answer:
For the first terms
Let the first term be and the common ratio is
.
… … … … … i)
For next terms
First term will be th and the common ratio is
.
… … … … … ii)
Hence proved.
Question 16: If are the roots of
and
are the roots of
, where
form a G.P. Prove that
Answer:
Given are the roots of
… … … … … i) and
… … … … … ii)
Similarly, are the roots of
… … … … … iii) and
… … … … … iv)
Now are in G.P.
Let the common ratio be . Therefore
Substituting in i) & iii) we get
… … … … … v)
… … … … … vi)
Dividing vi) by v) we get
v) ,
and
Hence
Question 17: How many terms of the G.P.
are needed to give the sum
?
Answer:
We have
Question 18: A person has parents,
grandparents,
great grand parents, and so on. Find the number of his ancestors during the
generations proceeding his own.
Answer:
We have
Question 19: If are the sums of
terms of
G.P.s whose first term is
in each and common ratios are
respectively, then prove that
Answer:
Given are the sums of
terms of
G.P.s whose first term is
in each and common ratios are
respectively
For and
Similarly,
Now
RHS. Hence proved.
Question 20: A G.P. consists of even number of terms. If the sum of all the terms is times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Answer:
Let the first term be and the common ratio be
Let there be terms in the G.P.
Sum of all the terms ( Sum of the terms occupying the odd places)
Question 21: Let be the
term of the G.P. of positive numbers. Let
and
, such that
. Prove that the common ratio of the G.P. is
.
Answer:
Let the first term be and the common ratio be
… … … … … i)
… … … … … ii)
Question 22: Find the sum of terms of the series whose every even term is
times the term before it and every odd term is
times the term before it, the first term being unity.
Answer:
Let the given series be
Given