Test: Arithmetic & Geometric Mean (June 18) - JEE MCQ

Given: 

Arithmetic Mean = A, Geometric Mean = G; Harmonic Mean = H

Concept:

1. If A is the arithmetic mean of numbers a and b and is given by

2. If G is the geometric mean of the numbers a and b and is given by 

  G = √ab

3. If H is the Harmonic mean of numbers a and b and is given by

4. The relation between AM, GM, and HM

• G² = AH
• AM  ≥  GM  ≥  HM

Calculation:

Using the above concept we can say that

G² = AH

G is the geometric mean between A and H.

Explanation:

Geometric mean of a and b is √ab. So by given condition,

Concept:

If GM of series x_i = a, then 

• GM of λx_i = λa
• GM of xiⁿ = aⁿ

Calculation:

Given GM of x_i = 10

Then GM of xi⁴ = 10⁴ = 10000

Now GM of 3xi4 = 3 × 10000 = 30000

Formula used:

For n terms,  a₁, a₂, a₃, ……., a_n

Geometric mean = (a₁.a₂.a₃....an)1/n

Calculation:

g is the geometric mean of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

⇒ g = (2×4×8×16×32×64×128×256×512×1024)110  

⇒ g = (2 × 2² × 2³ × 2⁴ × 2⁵ × 2⁶ × 2⁷ × 2⁸ × 2⁹ × 2¹⁰⁾¹¹⁰

⇒ g = 25.5

⇒ 2⁵ < g < 2⁶

∴ 32 < g < 64 is correct.

Concept:

Geometric mean is obtained by multiplying 'n' numbers together and taking the nth root of the product.

• Geometric mean of numbers x₁, x₂, …….,xn  is given by, Geometric Mean = (x₁x₂…….xn )^1/n

Calculation:

Given numbers: - 9 and 4

We know, the geometric mean is obtained by multiplying 'n' numbers together and taking the nth root of the product

∴ The geometric mean of -9 and 4 = 

Concept:

GM or Geometric Mean is the mean value or the central term in the set of numbers in geometric progression.

Geometric mean of a geometric sequence with 'n' terms is computed as n^th root of the product of all the terms in sequence taken together.

Calculations:

Given sequence is 1, 2, 4, 8, ______, 2ⁿ

Rewrite the sequence, 2^0, 2¹, 2^2, ....2ⁿ

Hence, the geometric mean of the sequence 1, 2, 4, 8, ______, 2ⁿ is 2^n/2

Concept:

Geometric mean is obtained by multiplying 'n' numbers together and taking the nth root of the product.

Eg:- The geometric mean of numbers x₁, x₂, …….,x_n  is given by, Geometric Mean = (x₁x₂…….x_n )^1/n

Calculation:

Given numbers are 2, 0.2 and 0.02

According to the formula used

Geometric Mean of 2, 0.2 and 0.02 = (2×0.2×0.02)^1/3

⇒(0.008)^1/3

⇒ 0.2

Hence, option (4) is correct. 

Concept:

The geometric mean of the numbers a₁, a₂,...,a_n is given by  ⇔ G = (a₁×a₂⋯×a_n)^1/n

Calculation:

GIven data: 2, 4, 8,16, 32?

Geometric mean = (2×4×8×16×32)^1/5

= (2¹ × 2² × 2³ × 2⁴ × 2⁵)^1/5

=(215)15

= 2³

= 8

Hence, option (3) is correct.

Concept:

The geometric mean is defined as the nth root of the product of n numbers.

The geometric mean of a data set {a₁,a₂,…,a_n} is given by: 

GM={a₁ × a₂×…× a_n}1/n

Calculation:

To Find: Geometric mean of 6, 8, 16 and 27

Here n = 4

Now,

CONCEPT:

If a, b and c are in a GP then b² = ac

CALCULATION:

Given: The numbers - 2/7, x, - 7/2 are in GP

As we know that, if a, b and c are in GP then b2 = ac

Here, a = - 2/7, b = x and c = - 7/2

⇒ x² = (-2/7) × (-7/2) = 1

⇒ x = ± 1

Hence, correct option is 3.