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Sequences & Series: Let Represent Arithmetic Means Represent Geometric Means Rep

JEE Maths question with a full step-by-step solution.

Question

Let

A_i

(i = 1, 2, \ldots, 6)

represent

6

arithmetic means,

G_i

(i = 1, 2, \ldots, 6)

represent

6

geometric means and

H_i

(i = 1, 2, \ldots, 6)

represent

6

harmonic means between the two numbers

2

and

72

, and let

G

represent the single geometric mean between

2

and

72

. Then the value of

\frac{\left(\displaystyle\prod_{i=1}^{6} G_i\right)\left(\displaystyle\sum_{i=1}^{6} \frac{1}{H_i}\right)} {\displaystyle\sum_{i=1}^{6} A_i}

G

G^{2}

G^{4}

correct

G^{6}

Solution

Step 1: Find

G

first, since the options are powers of it.

G = \sqrt{2 \times 72} = \sqrt{144} = 12 .

Step 2: Handle

\sum A_i

. Inserting

6

arithmetic means makes

2,\ A_1,\ A_2,\ \ldots,\ A_6,\ 72

an AP of

8

terms. For any AP, terms equidistant from the ends have the same sum, so pairing

A_1

with

A_6

A_2

with

A_5

A_3

with

A_4

gives each pair the sum

2 + 72 = 74

\sum_{i=1}^{6} A_i = 3 \times 74 = 222 .

Step 3: Handle

\prod G_i

. Inserting

6

geometric means makes

2, G_1, \ldots, G_6, 72

a GP, and terms equidistant from the ends have the same product

2 \times 72 = 144

\prod_{i=1}^{6} G_i = 144^{3} = \left(12^{2}\right)^{3} = 12^{6} .

Step 4: Handle

\sum \dfrac{1}{H_i}

. The defining property of harmonic means is that the reciprocals form an AP, so

\dfrac12, \dfrac{1}{H_1}, \ldots, \dfrac{1}{H_6}, \dfrac{1}{72}

is an AP. Pairing as in Step 2, each pair sums to

\dfrac12 + \dfrac{1}{72} = \dfrac{36+1}{72} = \dfrac{37}{72}

\sum_{i=1}^{6} \frac{1}{H_i} = 3 \times \frac{37}{72} = \frac{37}{24} .

Step 5: Put the three pieces together.

\frac{12^{6} \times \dfrac{37}{24}}{222} = \frac{12^{6} \times 37}{24 \times 222} .

Step 6: Simplify the numbers. Since

222 = 6 \times 37

, the

37

cancels:

\frac{12^{6}}{24 \times 6} = \frac{12^{6}}{144} = \frac{12^{6}}{12^{2}} = 12^{4} .

Step 7: Express in terms of

G = 12

12^{4} = G^{4} .

Answer: (3).

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