IB Maths AI HL Logs and Infinite Geometric Series Questions

1. Identify the formula for the sum to infinity of a geometric series: $S_{\mathrm{\infty }}=\frac{u_1}{1-r}$.

2. Substitute the known values into the formula: $S_{\mathrm{\infty }}=\frac{45}{1-0.92}$.

3. Simplify the denominator: $1-0.92=0.08$.

4. Evaluate the final division: $S_{\mathrm{\infty }}=\frac{45}{0.08}$.

5. The exact sum of the infinite series is $562.5$.

1. Isolate the exponential term by dividing both sides by $200$: $e^{-0.05t}=\frac{40}{200}=0.2$.

2. Rewrite $0.2$ as a fraction to help with logarithm laws: $e^{-0.05t}=\frac{1}{5}=5^{-1}$.

3. Take the natural logarithm ($\ln$) of both sides to remove the base $e$: $\ln (e^{-0.05t})=\ln (5^{-1})$.

4. Simplify using power laws: $-0.05t=-\ln 5$.

5. Divide to isolate $t$: $t=\frac{-\ln 5}{-0.05}=$ $20\ln 5$.

1. Rewrite the terms using the power law for logarithms ($\ln (a^b)=b\ln a$): $u_1=\ln 2$, $u_2=2\ln 2$, and $u_3=3\ln 2$.

2. Calculate the difference between consecutive terms: $u_2-u_1=\ln 2$ and $u_3-u_2=\ln 2$. Since the difference is constant, the sequence is arithmetic with $d=\ln 2$.

3. Identify the arithmetic series sum formula: $S_n=\frac{n}{2}(2u_1+(n-1)d)$.

4. Substitute $n=10,u_1=\ln 2$, and $d=\ln 2$: $S_{10}=5(2\ln 2+9\ln 2)$.

5. Simplify the brackets to find the exact sum: $S_{10}=5(11\ln 2)=$ $55\ln 2$.