Identify the first term and common ratio.

Question

Identify the first term and common ratio.

Asked on March 27, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-27T02:29:31.140Z

Step 1: Identify the first term and common ratio. The sequence is a geometric progression: $36, 12, 4, \dots$ The first term is $a = 36$. The common ratio is $r = \frac{12}{36} = \frac{1}{3}$. Step 2: Calculate the sum to infinity. The sum to infinity of a geometric progression is given by the formula $S_{\infty} = \frac{a}{1-r}$, provided that $|r| < 1$. In this case, $|r| = \left|\frac{1}{3}\right| < 1$, so the sum to infinity exists. $$S_{\infty} = \frac{36}{1 - \frac{1}{3}}$$ $$S_{\infty} = \frac{36}{\frac{3}{3} - \frac{1}{3}}$$ $$S_{\infty} = \frac{36}{\frac{2}{3}}$$ $$S_{\infty} = 36 \times \frac{3}{2}$$ $$S_{\infty} = 18 \times 3$$ $$S_{\infty} = 54$$ The correct option is B. The sum to infinity is $\boxed{\text{54}}$. 27. Step 1: Calculate the printing rate. The computer prints 2100 words in 5 minutes. Rate = $\frac{\text{Number of words}}{\text{Time}}$ $$\text{Rate} = \frac{2100 \text{ words}}{5 \text{ minutes}} = 420 \text{ words/minute}$$ Step 2: Calculate the number of words printed in $6\frac{1}{2}$ minutes. Time = $6\frac{1}{2}$ minutes = $6.5$ minutes. Number of words = Rate $\times$ Time $$\text{Number of words} = 420 \text{ words/minute} \times 6.5 \text{ minutes}$$ $$\text{Number of words} = 420 \times \frac{13}{2}$$ $$\text{Number of words} = 210 \times 13$$ $$\text{Number of words} = 2730$$ The correct option is B. The number of words printed is $\boxed{\text{2730}}$. 28. The number given is 50.999. The question asks to express it correct to 3 decimal places. If 50.999 is interpreted as the repeating decimal $50.999\dots$, then it is equivalent to 51. To express 51 correct to 3 decimal places, we write it as 51.000. Among the given options, 51.0 (Option A) is equivalent to 51.000. The correct option is A. 50.999, expressed correct to 3 decimal places, is $\boxed{\text{51.0}}$. 29. The number is 4.035. We need to find the value of the digit 3. The digit 3 is in the hundredths place. Its value is $3 \times \frac{1}{100} = 0.03$. The correct option is A. The value of the digit 3 in 4.035 is $\boxed{\text{0.03}}$. 30. The symbol $\le$ means "less than or equal to". The correct option is D. The symbol $\le$ means $\boxed{\text{Less than or equal to}}$. 31. Step 1: Locate point T on the graph. By observing the coordinates on the graph, point T is located at $x = -2$ and $y = -2$. The coordinates of point T are $(-2, -2)$. The correct option is B. The point T on the graph is $\boxed{\text{(-2, -2)}}$. 32. Step 1: Apply the order of operations (PEMDAS/BODMAS). Multiplication should be performed before subtraction. $$16 - 2 \times 3$$ $$16 - 6$$ $$10$$ The correct option is B. Evaluating $16 - 2 \times 3$ gives $\boxed{\text{10}}$. 33. Step 1: Simplify the equation of the line. The equation of the line is $2y = 4x$. Divide both sides by 2: $$y = 2x$$ Step 2: Check which option satisfies the equation $y = 2x$. A (4, 2): $2 = 2(4) \Rightarrow 2 = 8$ (False) B (1, 2): $2 = 2(1) \Rightarrow 2 = 2$ (True) C (2, 1): $1 = 2(2) \Rightarrow 1 = 4$ (False) D (1, 1): $1 = 2(1) \Rightarrow 1 = 2$ (False) The point (1

Accepted Answer · 2026-03-27T02:29:31.140Z

Step 1: Identify the first term and common ratio. The sequence is a geometric progression: $36, 12, 4, \dots$ The first term is $a = 36$. The common ratio is $r = \frac{12}{36} = \frac{1}{3}$. Step 2: Calculate the sum to infinity. The sum to infinity of a geometric progression is given by the formula $S_{\infty} = \frac{a}{1-r}$, provided that $|r| < 1$. In this case, $|r| = \left|\frac{1}{3}\right| < 1$, so the sum to infinity exists. $$S_{\infty} = \frac{36}{1 - \frac{1}{3}}$$ $$S_{\infty} = \frac{36}{\frac{3}{3} - \frac{1}{3}}$$ $$S_{\infty} = \frac{36}{\frac{2}{3}}$$ $$S_{\infty} = 36 \times \frac{3}{2}$$ $$S_{\infty} = 18 \times 3$$ $$S_{\infty} = 54$$ The correct option is B. The sum to infinity is $\boxed{\text{54}}$. 27. Step 1: Calculate the printing rate. The computer prints 2100 words in 5 minutes. Rate = $\frac{\text{Number of words}}{\text{Time}}$ $$\text{Rate} = \frac{2100 \text{ words}}{5 \text{ minutes}} = 420 \text{ words/minute}$$ Step 2: Calculate the number of words printed in $6\frac{1}{2}$ minutes. Time = $6\frac{1}{2}$ minutes = $6.5$ minutes. Number of words = Rate $\times$ Time $$\text{Number of words} = 420 \text{ words/minute} \times 6.5 \text{ minutes}$$ $$\text{Number of words} = 420 \times \frac{13}{2}$$ $$\text{Number of words} = 210 \times 13$$ $$\text{Number of words} = 2730$$ The correct option is B. The number of words printed is $\boxed{\text{2730}}$. 28. The number given is 50.999. The question asks to express it correct to 3 decimal places. If 50.999 is interpreted as the repeating decimal $50.999\dots$, then it is equivalent to 51. To express 51 correct to 3 decimal places, we write it as 51.000. Among the given options, 51.0 (Option A) is equivalent to 51.000. The correct option is A. 50.999, expressed correct to 3 decimal places, is $\boxed{\text{51.0}}$. 29. The number is 4.035. We need to find the value of the digit 3. The digit 3 is in the hundredths place. Its value is $3 \times \frac{1}{100} = 0.03$. The correct option is A. The value of the digit 3 in 4.035 is $\boxed{\text{0.03}}$. 30. The symbol $\le$ means "less than or equal to". The correct option is D. The symbol $\le$ means $\boxed{\text{Less than or equal to}}$. 31. Step 1: Locate point T on the graph. By observing the coordinates on the graph, point T is located at $x = -2$ and $y = -2$. The coordinates of point T are $(-2, -2)$. The correct option is B. The point T on the graph is $\boxed{\text{(-2, -2)}}$. 32. Step 1: Apply the order of operations (PEMDAS/BODMAS). Multiplication should be performed before subtraction. $$16 - 2 \times 3$$ $$16 - 6$$ $$10$$ The correct option is B. Evaluating $16 - 2 \times 3$ gives $\boxed{\text{10}}$. 33. Step 1: Simplify the equation of the line. The equation of the line is $2y = 4x$. Divide both sides by 2: $$y = 2x$$ Step 2: Check which option satisfies the equation $y = 2x$. A (4, 2): $2 = 2(4) \Rightarrow 2 = 8$ (False) B (1, 2): $2 = 2(1) \Rightarrow 2 = 2$ (True) C (2, 1): $1 = 2(2) \Rightarrow 1 = 4$ (False) D (1, 1): $1 = 2(1) \Rightarrow 1 = 2$ (False) The point (1