$\"\"$

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The compound investment is $\"\"$ and annual rate is $\"\"$ .

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The annual rate $\"\"$ .

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The amount after maturity period is the double to the investment is $\"\"$ .

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The formula for $\"\"$ compounding per year: $\"\"$ . $\"\"$

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(a)

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Annually compounding means $\"\"$ .

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Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Take natural $\"\"$ of each side.

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$\"\"$

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Apply inverse property: $\"\"$ .

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$\"\"$

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Apply change-of-base formula : $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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The required time to double the investment is $\"\"$ .

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$\"\"$

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(b)

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Monthly compounding means $\"\"$ .

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Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Take $\"\"$ of each side.

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$\"\"$

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Apply inverse property: $\"\"$ .

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$\"\"$

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Apply change-of-base formula : $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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The required time is $\"\"$ for monthly.

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$\"\"$

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(c)

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daily compounding means $\"\"$ .

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Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Take $\"\"$ of each side.

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$\"\"$

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Apply inverse property: $\"\"$ .

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$\"\"$

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Apply change-of-base formula : $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ \ \

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$\"\"$

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$\"\"$ .

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The required time is $\"\"$ for daily.

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$\"\"$

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(d)

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The formula for continuous compounding: $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Take $\"\"$ on each side.

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$\"\"$

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Apply inverse property: $\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The required time is $\"\"$ for continuously.

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$\"\"$

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(a) The required time is $\"\"$ for annually,

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(b) The required time is $\"\"$ for monthly,

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(c) The required time is $\"\"$ for daily, and

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(d) The required time is $\"\"$ for continuously.