Janette invested 2000 $a t$ 5\% $c o m p o u n d e d a n n u a l l y f o r$ 5 $y e a r s . H o w m u c h i n t e r e s t d i d s h e e a r n a t t h e e n d o f t h e$ 5$ year period?

552.56$

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952.56$

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352.56$

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652.56$

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Solution

The correct option is A 552.56$
In the first year, she has the principal plus the interest on the principal.
$\Rightarrow$ $P_{1} = 2000 + (5 % \times 2000) = 2000 (1 + 5 %)$
In the second year, she has the principal $P_{1}$ plus the interest on $P_{1}$
$P_{2} = P_{1} + (5 % \times P_{1}) = P_{1} (1 + 5 %)$
Substitute $P_{1}$ by $2000 (1 + 5 %)$ found above to find
$P_{2} = 2000 \times (1 + 5 %)^{2}$
Continuing with this process, it can easily be shown that a end of the $5^{t h}$ year, the principal is given by
$P_{5} = 2000 \times (1 + 5 %)^{5}$
$= 2000 \times (1 + 0.05)$
$= $ 2552.56$
The interest earned at the end of the $5$ years $= $ (2552.56 - 2000) = $ 552.56$

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