QCAA General Mathematics Growth and decay in sequences

This value results from an incorrect calculation of the bounce heights or their difference.

This is the height of the second bounce ($t_2 = 120$) or the difference between the first and second bounces, rather than the difference between the first and third.

This error likely occurs if the formula $240 \times 0.5^n$ is used instead of $240 \times 0.5^{(n-1)}$, yielding a third bounce of 30 and a difference of $240 - 30 = 210$.

This is incorrect because the difference between terms is not constant. This is actually a geometric sequence where each term is multiplied by a common ratio of $-2$.

This is incorrect because the difference between consecutive terms changes from $-10.5$ to $17.5$. An arithmetic sequence must have a constant common difference.

This is incorrect because the difference between terms is not constant ($7.0 - 3.5 = 3.5$, but $14.0 - 7.0 = 7.0$). This represents a geometric sequence with a common ratio of $2$.

This is incorrect because it represents the height of the ball after the fourth bounce ($25.6 \times 0.75^4 = 8.1$ m), not the third.

This is incorrect because it represents the height of the ball after the second bounce ($25.6 \times 0.75^2 = 14.4$ m).

This is incorrect because it represents the height of the ball after the first bounce ($25.6 \times 0.75 = 19.2$ m).

This is the first term of the sequence ($a_1$), not the common difference between terms.

This value is obtained by subtracting the second term from the first ($4 - 2$), but the formula for common difference is $a_{n+1} - a_n$ (second term minus first term).

This is the fifth term of the sequence, not the constant value added to each term to get the next.

This option correctly identifies the geometric nature of the growth, but the value for $n$ is incorrect. Since January 2022 is the 1st term ($n=1$), January 2030 is 8 years later, making it the 9th term ($n=9$).

This option uses the arithmetic sequence formula, which models adding a fixed amount ($d$) each year. Since the population doubles (multiplies), a geometric formula is required.

This is incorrect because it applies an arithmetic rule ($t_n = t_1 + (n-1)d$). Doubling represents exponential growth, which must be modeled by a geometric sequence.

Incorrect. A 10% reducing balance rate would leave a final value of $3000(1 - 0.10)^4 = ＄1968.30$, which is much higher than the actual final value of ＄600.

Incorrect. Applying a 15% reducing balance rate gives a final value of $3000(1 - 0.15)^4 = ＄1566.02$, which does not match the target of ＄600.

Incorrect. This represents the straight-line depreciation rate, where the computer loses ＄600 per year (20% of ＄3000), rather than the reducing balance rate.

Incorrect. A 25% reducing balance rate results in a final value of $3000(1 - 0.25)^4 = ＄949.22$, which is still higher than ＄600.

Incorrect. This formula calculates compound interest ($A = P(1 + r)^t$), but the question specifies simple interest.

Incorrect. This calculates three times the total balance after one year, which incorrectly triples the initial principal instead of just multiplying the interest by 3.

Incorrect. This cubes the interest earned in a single year instead of multiplying it by the number of years ($t=3$).

This incorrect answer ignores the negative sign of the common ratio. Since $r = -0.9$ and the exponent is odd ($n-1=3$), the 4th term must be negative.

This option incorrectly treats the sequence as arithmetic (subtracting 100 each time) rather than geometric, and ignores the negative sign of the second term.

This answer suggests an arithmetic pattern rather than a geometric one. In a geometric sequence, terms are multiplied by a common ratio, not added or subtracted by a constant.