Part 1Consider a twominus−year investment: Given a constant and positive interest rate, the interest earned in the second year will be greater than the interest earned in the first year (assuming annual compounding). Part 1
True False

Question

Part 1Consider a twominus−year ​investment: Given a constant and positive interest​ rate, the interest earned in the second year will be greater than the interest earned in the first year​ (assuming annual​ compounding). Part 1
True False

BlackTom AI · Accepted Answer

The question asks us to assess whether, for a two-year investment with a constant positive interest rate and annual compounding, the interest earned in the second year is greater than the interest earned in the first year.
Option 1: True. In year 1, the interest earned is P × r, where P is the principal and r is the annual rate. In year 2, the amount from year 1 grows to P × (1 + r), so the interest earned in year 2 is [P × (1 + r) − P] = P × r × (1 + r). Since r > 0, we have (1 + r) > 1, which means P × r × (1 + r) > P × r. Therefore, the second-year interest is indeed greater than the first-year interest.
Option 2: False. If we suppose False, we would imply that the second year’s interest is not greater than the first year’s interest. But as shown above, with a positive r, the growth factor (1 + r) amplifies the interest earned in the first year, making the second-year interest larger. This contradicts the claimed False statement, so this option mischaracterizes the scenario.
Additional nuance: The comparison hinges on the effect of compounding. With annual compounding and a positive rate, each successive year's interest is computed on an elevated principal, ensuring the sequence of year-by-year interest amounts is increasing. If the rate were zero or negative, the conclusion would differ, but the premise specifies a positive rate, so the logic above applies.
In summary, the claim in the True option aligns with the mathematics of simple-year growth under compound interest, while the False option contradicts the immediate effect of compounding at a positive rate.

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Part 1Consider a twominus−year investment: Given a constant and positive interest rate, the interest earned in the second year will be greater than the interest earned in the first year (assuming annual compounding). Part 1 True False

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