*Notes: *

Question 1: Find the sum of the following geometric progressions:

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Question 2: Find the sum of the following geometric progressions:

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Question 3: Evaluate the following:

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Question 4: Find the sum of the following series:

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Question 6: How many terms of the series must be taken to make the sum equal to ?

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Question 7: How many terms of the sequence must be taken to make the sum ?

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Question 8: The sum of of the G.P. is is 381. Find the value of .

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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:

We k

… … … … … 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:

and the common ratio is

Let

or

. But

Answer:

… … … … … i)

… … … … … ii)

Dividing ii) by i) we get

Substituting in i) we get

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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:

… … … … … i)

Similarly, … … … … … ii)

Dividing ii) by i) we get

Question 14: If be respectively the sums of of a G.P., then prove that .

Answer:

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 term is .

Answer:

For the first

and the common ratio is .

… … … … … i)

For next

First term will be th and the common ratio is .

… … … … … ii)

Hence proved.

Question 16: If are the roots of are the roots of , w form a G.P. Prove that

Answer:

are the roots of

… … … … … … … … … … ii)

Similarly, are the roots of

… … … … … … … … … … iv)

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) ,

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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.

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Question 19: If are the sums of of G.P.s whose first term is in each and common ratios are respectively, then prove that

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are the sums of of G.P.s whose first term is in each and common ratios are respectively

Similarly,

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 there be in the G.P.

Sum of all the terms ( Sum of the terms occupying the odd places)

Answer:

… … … … … i)

… … … … … ii)

Question 22: Find the sum of 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.

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