*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