Geometric Series Worksheet

Geometric Series Worksheet
  • Page 1
 1.  
If a, b and c are in G.P as well as in A.P then
a.
a = bc
b.
abc
c.
ab = c
d.
a = b = c


Solution:

Given a, b, c are in G.P. So, b2 = ac ---------- (1)

Also given a, b, c are in A.P. So, b = a+c2 ---------- (2)

b2 = (a+c)24
[Square the equation (2) on both sides.]

(a+c)24 = ac
[Equate the values of b2, Use steps 1 and 3.]

(a + c)2 = 4ac
(a + c)2 - 4ac = 0
[Simplify.]

a2 + c2 + 2ac - 4ac = 0
(a - c)2 = 0
[a2 + c2 - 2ac = (a - c)2.]

(a - c) = 0, a = c
[Simplify.]

b = a+c2 =c+c2 = c
[Substitute the value of a in equation (2) to find b.]

Therefore, a = b = c
[From steps 7 & 8.]


Correct answer : (4)
 2.  
The value of 1413141914127 ...................... ∞
a.
14
b.
1
c.
1 14
d.
14


Solution:

1413141914127 ...................... ∞
[As per the question.]

= 1413+19+127...................... 
[Use the formula: aman =am+n.]

= 1413(1-13)
[Use the formula: sum of infinite terms in geometric series = a1 - r.]

= 141323

= 1412

= 14


Correct answer : (1)
 3.  
If the common ratio of a GP is - 8 25 and the sum to infinity is 1625 33, then find the first term.
a.
99
b.
65
c.
98
d.
67


Solution:

a1-r = 1625 / 33
[Formula for sum of infinite geometric series is a1-r where a is the first term and r is the common ratio.]

a1+825 = 1625 / 33
[Substitute r = - 8 / 25.]

a3325 = 1625 / 33
[Simplify.]

a = 1625 / 33× (33 / 25)
[Simplify.]

a = 65

Therefore, the first term = 65.


Correct answer : (2)
 4.  
If a1x=b1y=c1z and a, b, c are in G.P then x, y, z are in ______
a.
None of these
b.
A.G.P
c.
G.P
d.
A.P


Solution:

Let a1x=b1y=c1z = k.

a = kx, b = ky, c = kz
[If a1x = k then a = kx.]

Given that a, b, c are in G.P so, b2 = ac.

k2y = kx. kz
[Substitute in step 3.]

k2y = kx + z
[When bases are equal powers should be added.]

2y = x + z
[Since bases on either sides are same, exponents should be equated.]

Therefore x, y, z are in A.P.


Correct answer : (4)
 5.  
What is the sum of the first 6 terms of the geometric series 1 + 4 + 16 + ... ?
a.
16
b.
1365
c.
4
d.
1024


Answer: (b)


Correct answer : (2)
 6.  
Find the sum of 10 terms, S10 in the geometric series:
1 + 1 3 + 1 9 + 1 27 + ...
a.
32(1 - 1310)
b.
13(1 - 1310)
c.
23(1 - 1310)
d.
139


Answer: (a)


Correct answer : (1)
 7.  
Find the sum of the infinite geometric series:
1 + 1 9 + 1 81 + ......
a.
1 9
b.
9 8
c.
9 10
d.
8 9


Answer: (b)


Correct answer : (2)
 8.  
How many terms of the geometric sequence 7, 72, 73... are required to obtain a sum of 2800?
a.
4
b.
3
c.
6
d.
5


Answer: (a)


Correct answer : (1)
 9.  
Find the common ratio of the geometric series, whose sum of first 39 terms, S39 = 1039 - 19.
a.
9
b.
39
c.
1
d.
10


Answer: (d)


Correct answer : (4)
 10.  
The sum of an infinite geometric series is 352 andthe first term of the sequence is 32. Find the common ratio of the series.
a.
1 10
b.
11 10
c.
10 11
d.
1 11


Answer: (c)


Correct answer : (3)

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