Quarterly growth written as a function of years

Two pieces go into the model.

Growth factor. Growing by 6.1% means each quarter's value is the previous value plus 0.061 of it: $1+0.061=1.061$. The factor is 1.061, not 0.061.

Exponent. The factor applies once per quarter, and there are 4 quarters in a year. After $t$ years, $4t$ quarters have passed, so the factor is applied $4t$ times.

With the starting value of 54 million dollars: $$M(t)=54(1.061{)}^{4t}$$ That is choice D.

A check: after 1 year, $t=1$ gives $54(1.061{)}^4$, four quarters of growth, which is what the situation describes.

Each choice

• A. Using the rate 0.061 as the factor would shrink the value to about 6% of itself each period. The exponent $\frac{t}{4}$ is also upside down: it counts one growth period every four years.
• B. Here 0.061 is the rate, not the factor. Multiplying by 0.061 every quarter models a steep decline, not 6.1% growth. The factor is $1+0.061$.
• C. The factor is right but the exponent is upside down. $\frac{t}{4}$ means one growth step every four years. The number of quarters in $t$ years is $4t$.
• D (correct). The factor 1.061 is applied once per quarter, and $t$ years contain $4t$ quarters.