$\"\"$

\

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

\

The annual rate $\"\"$ .

\

The amount after maturity period is the double to the investment

\

is $\"\"$ .

\

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

\

(a)

\

Annually compounding means $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

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

\

$\"\"$

\

Apply inverse property: $\"\"$ .

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$ \ \

\

(b)

\

Monthly compounding means $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Take $\"\"$ of each side.

\

$\"\"$

\

Apply inverse property: $\"\"$ .

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

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

\

$\"\"$ \ \

\

(c)

\

Daily compounding means $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Take $\"\"$ of each side.

\

$\"\"$

\

Apply inverse property: $\"\"$ .

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$

\

(d)

\

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

\

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

\

$\"\"$

\

$\"\"$

\

Take $\"\"$ on each side.

\

$\"\"$

\

Apply inverse property: $\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$

\

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

\

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

\

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

\

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