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2 years ago

Explicit formula of 400, 100, 25, 25/4 and use it to find the 15th term

Mathematics

1 answer:

erastovalidia [21]2 years ago

7 0

Answer:

The 15th term is 25/16'777,216

Step-by-step explanation:

Aₙ = Aₙ₋₁ ÷ 4; A₁ = 400

A₂ = A₁ ÷ 4 = 400 ÷ 4 = 100

A₃ = A₂÷ 4 = 100 ÷ 4 = 25

A₄ = A₃ ÷ 4 = 25 ÷ 4 = 25/4

A₅ = A₄÷ 4 = 25/4 ÷ 4 = 25/4 * 1/4 = 25/16

A₆ = A₅÷ 4 = 25/16 ÷ 4 = 25/16 * 1/4 = 25/64

A₇ = A₆÷ 4 = 25/64 ÷ 4 = 25/64 * 1/4 = 25/256

A₈ = A₇ ÷ 4 = 25/256 ÷ 4 = 25/256 * 1/4 = 25/1,024

A₉ = A₈÷ 4 = 25/1,024 ÷ 4 = 25/1,024 * 1/4 = 25/4,096

A₁₀ = A₉÷ 4 = 25/4,096 ÷ 4 = 25/4,096 * 1/4 = 25/16,384

A₁₁ = A₁₀ ÷ 4 = 25/16,384 ÷ 4 = 25/16,384 * 1/4 = 25/65,536

A₁₂ = A₁₁ ÷ 4 = 25/65,536 ÷ 4 = 25/65,536 * 1/4 = 25/262,144

A₁₃ = A₁₂ ÷ 4 = 25/262,144 ÷ 4 = 25/262,144 * 1/4 = 25/1'048,576

A₁₄ = A₁₃ ÷ 4 = 25/1'048,576 ÷ 4 = 25/1'048,576 * 1/4 = 25/4'194,304

A₁₅ = A₁₄ ÷ 4 = 25/4'194,304 ÷ 4 = 25/4'194,304 * 1/4 = 25/16'777,216

The 15th term is 25/16'777,216

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We have been given that a handbag contains five coins, four keys,

and eight mints.

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= 5 coins + 4 keys + 8 mints

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So, the total number of items in a handbag = 17

Here, two items are taken out of the handbag one after the other and not replaced.

We need to find the expected number of mints taken out of the handbag.

Number of items taken out = 2

Let x be the number of mints taken out of the handbag.

x can be 0, 1, or 2

For x = 0,

the two items taken out could be either coins or keys.

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So, the 1st pick = 9/17

and 2nd pick = 8/16

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For x = 1,

Assume that in the 1st pick mint is taken out.

So, 1st pick = 8/17

For second pick, it can be either coin or key.

This means, there are two possibilities for the second pick.

so, 2nd pick = 2 × 9/16

⇒ P(x = 1) = 8/17 × 9/16 × 2

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1st pick = 8/17

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⇒ P(x = 2) = 8/17 × 7/16

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So, the expected number of mints taken out of the handbag would be,

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Therefore, the expected number of mints taken out of the handbag = 0.94

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