Step 1:

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The compound investment $\"\"$ , annual rate $\"\"$ .

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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 n compounding per year: $\"\"$ .

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Step 2:

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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 is $\"\"$ .

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Step 3:

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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 $\"\"$ years for monthly.

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Step 4:

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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 $\"\"$ years for daily.

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Step 5:

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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 $\"\"$ years for continuously.

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Solution :

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(a) The required time is 7.3 years for annually.

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(b) The required time is 6.98 years for monthly.

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(c) The required time is 6.93 years for daily, and

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(d) The required time is 6.93 years for continuously.