Let's solve each question step by step:

12. Find the sum.

$\sum_{k=1}^{6} (k^2 - 3)$

This involves evaluating each term and summing them up.

Steps:

• Calculate $k^2 - 3$ for $k = 1, 2, 3, 4, 5, 6$:
  • $k = 1: 1^2 - 3 = 1 - 3 = -2$
  • $k = 2: 2^2 - 3 = 4 - 3 = 1$
  • $k = 3: 3^2 - 3 = 9 - 3 = 6$
  • $k = 4: 4^2 - 3 = 16 - 3 = 13$
  • $k = 5: 5^2 - 3 = 25 - 3 = 22$
  • $k = 6: 6^2 - 3 = 36 - 3 = 33$

Sum the values: $-2 + 1 + 6 + 13 + 22 + 33 = 73$

Answer: $\boxed{73}$

13. Find the sum of an arithmetic sequence.

$\sum_{k=0}^{5} (1 + 2.3k)$

Steps:

1. Calculate the first few terms:

   • $k = 0: 1 + 2.3(0) = 1$
   • $k = 1: 1 + 2.3(1) = 3.3$
   • $k = 2: 1 + 2.3(2) = 5.6$
   • $k = 3: 1 + 2.3(3) = 7.9$
   • $k = 4: 1 + 2.3(4) = 10.2$
   • $k = 5: 1 + 2.3(5) = 12.5$

2. Sum the terms: $1 + 3.3 + 5.6 + 7.9 + 10.2 + 12.5 = 40.5$

Answer: $\boxed{40.5}$

14. Find the sum of the infinite geometric series.

$1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots$

This is an infinite geometric series with:

• First term ($a$) = $1$,
• Common ratio ($r$) = $\frac{1}{3}$.

The formula for the sum of an infinite geometric series is: $S = \frac{a}{1 - r}$

Substitute the values: $S = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}$

Answer: $\boxed{\frac{3}{2} \text{ or } 1.5}$

15. Population growth problem.

$P_n = 45,000 (1.05)^n$

(a) Find the first five terms of the sequence.

Substitute $n = 0, 1, 2, 3, 4$:

• $P_0 = 45,000 (1.05)^0 = 45,000$
• $P_1 = 45,000 (1.05)^1 = 45,000 \times 1.05 = 47,250$
• $P_2 = 45,000 (1.05)^2 = 45,000 \times 1.1025 = 49,612.5 \approx 49,613$
• $P_3 = 45,000 (1.05)^3 = 45,000 \times 1.157625 = 52,093.125 \approx 52,093$
• $P_4 = 45,000 (1.05)^4 = 45,000 \times 1.21550625 = 54,697.78125 \approx 54,698$

Answers: $P_1 = 47,250,\ P_2 = 49,613,\ P_3 = 52,093,\ P_4 = 54,698$

(b) Find the population in 2014.

Since 2014 is 10 years after 2004, $n = 10$: $P_{10} = 45,000 (1.05)^{10}$ $P_{10} = 45,000 \times 1.628894626777442 = 73,300.258$

Rounded to the nearest whole number: $\boxed{73,300}$

Let me know if you'd like further details or have any questions! Here are 5 related questions to explore:

1. How do you simplify summations using properties?
2. What happens to a geometric series if the common ratio $|r| \geq 1$?
3. Can you calculate the growth rate for other exponential models?
4. How do you find the $n$-th term in an arithmetic sequence?
5. What is the difference between finite and infinite series?

Tip: For population growth, exponential models are useful for estimating future sizes accurately!