Sequences and Series Test

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Sequences and Series Test - 5

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Weekly Quiz Competition

Question 1
2 / -0.83

For a G.P. the ratio of the 7^th and the third terms is 16. The sum of 9 terms is 2555. What is the first term?

A
7

B
5

C
4

D
9

Solution

ar⁶ /ar² = 16
r⁴ = 16
r⁴ = (2)⁴
r = 2
S₉ = a(rⁿ - 1)
2555 = a((2)⁹ - 1)
2555 = a(512-1)
2555/511 = a
a = 5
Question 2
2 / -0.83

The G.M. of 5 and 8 is

A
40

B
2 √10

C
10 √2

D
20

Solution

The geometric mean of two numbers, say x, and y is the square root of their product x * y.
Mean = [x * y]^1/2
= [5 * 8]^1/2
= [40]^1/2
= 2(10)^1/2
Question 3
2 / -0.83

The 10^th term of an G.P. is and the 5th term is.What is the Common Ratio?

A
1/4

B
1/16

C
1/2

D
1/8

Solution

ar⁹ = 1/(2)⁸ -----------------------(1)
ar³ = 1/(2)² ------------------------------(2)
Comparing eq (1) &(2)
1/(2)⁸ r⁹ = 1/(2)^2 r⁴
1/r⁵ = (2)⁵
r⁵ = 1/(2)⁵
r = 1/2
Question 4
2 / -0.83

A sequence a₁ , a₂ , a₃ ,…, a_n is called ______ progression, if each term is non-zero and = constant for

A
geometric, k ≥1

B
arithmetic, k >1

C
geometric, k >1

D
arithmetic, k ≥1

Solution

The expression depicts the ratio of two numbers, if the ratio between the numbers is constant, then it will definately form a GP.
Question 5
2 / -0.83

Which of the following sequeces in GP will have common ratio 3,where n is an Integer?

A
g_n = 6(3^n-1 )

B
g_n = 3n² + 3n

C
g_n = 2n² + 3

D
g_n = 2n² + 3n

Solution

g_n = 6( 3^n-1 ) it is a geometric expression with coefficient of constant as 3^n-1 .So it is GP with common ratio 3.
Question 6
2 / -0.83

The third term of a G.P. is 3, the product of first five terms of this progression is:

A
81

B
27

C
243

D
none of these

Solution
Question 7
2 / -0.83

The sum of n terms of the sequence 8, 88, 888,…. is:

A

B

C

D

Solution

8(1) + 8(11) + 8(111) + 8(1111) +........up to n times
= 8{ 1 + 11 + 111 + 1111 + ....... up to n times}
= 8/9{9 + 99 + 999 + 9999 + ..... up to n times}
= 8/9 {(10-1 )+ (100-1) + (1000-1) + (10000-1) + ........up to n times}
= 8/9 {(10 + 100 + 1000 + 10000 +...... n times) - (n ×1)}
= 8/9{( (10 ×(10ⁿ -1))/(10-1)) -n}
= 80/81(10ⁿ - 1) - n
= 8/81(10ⁿ⁺¹ - 10 -9n)
Question 8
2 / -0.83

Which term of the following sequence is 64 ?
2 , 2 √2, 4, .....

A
12

B
10

C
11

D
8

Solution

Given sequence : 2, 2 √2, 4 ….
First term a₁ = a = 2 and 2nd term a₂ = 2 √2, then
Common ratio r = a₂ /a = (2 √2)/2
Let a_n = 64
∴ar^(n-1) = 64
⇒2.(√2)^(n-1) = 32
⇒(2)^(n-1)/2 = 32
∴(2)^(n-1)/2 = (2)⁵
⇒(n −1)/2 = 5,
⇒n = 11
Question 9
2 / -0.83

How many terms of the G.P. 4 + 16 + 64 + …will make the sum 5460?

A
9

B
7

C
8

D
6

Solution

Sum (S_n ) = a x (rⁿ -1)/(r-1)
5460 = 4 x (4ⁿ -1)/3
16380 = 4ⁿ⁺¹ - 4
16384 = 4ⁿ⁺¹
4⁷ = 4ⁿ⁺¹
7 = n + 1
n = 6
Question 10
2 / -0.83

What is the 50^th term of the sequence √3, 3, 3 √3, 9, ......

A
(√3)⁴⁹

B
(√3)⁵⁰

C
3⁴⁹

D
3⁵⁰

Solution

an = ar^(n-1)
Given, a = √3, r = 3/√3
r = √3
a₅₀ = ar^(n-1)
= (√3)(√3)^(50-1)
= (√3)(√3)⁴⁹
= (√3)⁵⁰