Several years ago, an initial amount of $100 was invested in an account that has interest compounded yearly. No additional money was deposited in the account. The amount, in dollars, in the account can be determined by the expression \( 100(1.025)^5(1.05)^{t-5}\), where \(t\) is the number of years since the initial deposit and \(t>5\). What is a possible interpretation of \((1.025)^5\) in the expression?

Summary Submit Skip Question

We can expand the expression to inspect what's happening.

$$ 100(1.025)^5(1.05)^{t-5}$$ $$ = 100(1.025)(1.025)(1.025)(1.025)(1.025) \cdot (1.05)^{t-5 }$$

The initial amount is multiplied by 1.025 five times. This corresponds to a 2.5% increase five times. Because the interest is compounded yearly, it would make sense that the 2.5% rate corresponds to the annual percent increase.

When \(t>5\), such as on the 6^th year,

$$ 100(1.025)^t \cdot (1.05)^{t-5 }$$ $$ = 100(1.025)^t \cdot (1.05)^{6-5 }$$ $$ = 100\colorbox{aqua}{(1.025)(1.025)(1.025)(1.025)(1.025)}\colorbox{yellow}{(1.05)} $$

It appears that after 5 years, the interest rate increases to 5%.

Therefore, the best interpretation is that the interest rate was 2.5% only for the first 5 years.