Understanding the Sequence Formula

Oct 13, 2025 by windailys.com 98 views

1. Finding the First 5 Terms of a Sequence

In this section, we will explore the sequence given by the formula $$t_n = 1 + \frac{(-1)^n}{2n2}$$ and find the first 5 terms of the sequence.

Understanding the Sequence Formula

The sequence formula is given by $$t_n = 1 + \frac{(-1)^n}{2n2}$$. This formula indicates that each term of the sequence is determined by the value of n, where n is a positive integer. The term $$\frac{(-1)^n}{2n2}$$ represents the alternating pattern of the sequence, where the sign of the term alternates between positive and negative.

Finding the First 5 Terms of the Sequence

To find the first 5 terms of the sequence, we need to substitute the values of n into the sequence formula and calculate the corresponding terms.

For n = 1, we have $$t_1 = 1 + \frac{(-1)^1}{2(1)2} = 1 - \frac{1}{2} = \frac{1}{2}$$.
For n = 2, we have $$t_2 = 1 + \frac{(-1)^2}{2(2)2} = 1 + \frac{1}{8} = \frac{9}{8}$$.
For n = 3, we have $$t_3 = 1 + \frac{(-1)^3}{2(3)2} = 1 - \frac{1}{18} = \frac{17}{18}$$.
For n = 4, we have $$t_4 = 1 + \frac{(-1)^4}{2(4)2} = 1 + \frac{1}{32} = \frac{33}{32}$$.
For n = 5, we have $$t_5 = 1 + \frac{(-1)^5}{2(5)2} = 1 - \frac{1}{50} = \frac{49}{50}$$.

Therefore, the first 5 terms of the sequence are $$\frac{1}{2}, \frac{9}{8}, \frac{17}{18}, \frac{33}{32}, \frac{49}{50}$$.

2. Solving a Linear Function

In this section, we will explore the linear function given by the formula $$f(x) = 5x + 2$$ and solve the following problem:

(i) Find the Value of f(3)

To find the value of f(3), we need to substitute x = 3 into the function formula and calculate the corresponding value.

f(3) = 5(3) + 2 = 15 + 2 = 17$$. Therefore, the value of f(3) is 17. ### (ii) Find the Value of f(-2) To find the value of f(-2), we need to substitute x = -2 into the function formula and calculate the corresponding value. $$f(-2) = 5(-2) + 2 = -10 + 2 = -8$$. Therefore, the value of f(-2) is -8. ### (iii) Find the Value of f(0) To find the value of f(0), we need to substitute x = 0 into the function formula and calculate the corresponding value. $$f(0) = 5(0) + 2 = 0 + 2 = 2$$. Therefore, the value of f(0) is 2. ### (iv) Find the Value of f(a) To find the value of f(a), we need to substitute x = a into the function formula and calculate the corresponding value. $$f(a) = 5a + 2$$. Therefore, the value of f(a) is 5a + 2. ### (v) Find the Value of f(x + 1) To find the value of f(x + 1), we need to substitute x = x + 1 into the function formula and calculate the corresponding value. $$f(x + 1) = 5(x + 1) + 2 = 5x + 5 + 2 = 5x + 7$$. Therefore, the value of f(x + 1) is 5x + 7. ### (vi) Find the Value of f(x - 2) To find the value of f(x - 2), we need to substitute x = x - 2 into the function formula and calculate the corresponding value. $$f(x - 2) = 5(x - 2) + 2 = 5x - 10 + 2 = 5x - 8$$. Therefore, the value of f(x - 2) is 5x - 8. ### (vii) Find the Value of f(2x) To find the value of f(2x), we need to substitute x = 2x into the function formula and calculate the corresponding value. $$f(2x) = 5(2x) + 2 = 10x + 2$$. Therefore, the value of f(2x) is 10x + 2. ### (viii) Find the Value of f(3x - 1) To find the value of f(3x - 1), we need to substitute x = 3x - 1 into the function formula and calculate the corresponding value. $$f(3x - 1) = 5(3x - 1) + 2 = 15x - 5 + 2 = 15x - 3$$. Therefore, the value of f(3x - 1) is 15x - 3. ### (ix) Find the Value of f(x/2) To find the value of f(x/2), we need to substitute x = x/2 into the function formula and calculate the corresponding value. $$f(x/2) = 5(x/2) + 2 = 2.5x + 2$$. Therefore, the value of f(x/2) is 2.5x + 2. ### (x) Find the Value of f(2x + 1) To find the value of f(2x + 1), we need to substitute x = 2x + 1 into the function formula and calculate the corresponding value. $$f(2x + 1) = 5(2x + 1) + 2 = 10x + 5 + 2 = 10x + 7$$. Therefore, the value of f(2x + 1) is 10x + 7. ### (xi) Find the Value of f(3x + 2) To find the value of f(3x + 2), we need to substitute x = 3x + 2 into the function formula and calculate the corresponding value. $$f(3x + 2) = 5(3x + 2) + 2 = 15x + 10 + 2 = 15x + 12$$. Therefore, the value of f(3x + 2) is 15x + 12. ### (xii) Find the Value of f(x - 1/2) To find the value of f(x - 1/2), we need to substitute x = x - 1/2 into the function formula and calculate the corresponding value. $$f(x - 1/2) = 5(x - 1/2) + 2 = 5x - 5/2 + 2 = 5x - 1/2$$. Therefore, the value of f(x - 1/2) is 5x - 1/2. ### (xiii) Find the Value of f(2x - 1) To find the value of f(2x - 1), we need to substitute x = 2x - 1 into the function formula and calculate the corresponding value. $$f(2x - 1) = 5(2x - 1) + 2 = 10x - 5 + 2 = 10x - 3$$. Therefore, the value of f(2x - 1) is 10x - 3. ### (xiv) Find the Value of f(x/3) To find the value of f(x/3), we need to substitute x = x/3 into the function formula and calculate the corresponding value. $$f(x/3) = 5(x/3) + 2 = 5/3x + 2$$. Therefore, the value of f(x/3) is 5/3x + 2. ### (xv) Find the Value of f(3x + 1) To find the value of f(3x + 1), we need to substitute x = 3x + 1 into the function formula and calculate the corresponding value. $$f(3x + 1) = 5(3x + 1) + 2 = 15x + 5 + 2 = 15x + 7$$. Therefore, the value of f(3x + 1) is 15x + 7. ### (xvi) Find the Value of f(x - 3/2) To find the value of f(x - 3/2), we need to<br/> **Mathematics Q&A** ===================== **Q1: What is the formula for the sequence given by the formula $$t_n = 1 + \frac{(-1)^n}{2n^2}$$?** A1: The formula for the sequence is $$t_n = 1 + \frac{(-1)^n}{2n^2}$$. **Q2: How do you find the first 5 terms of the sequence?** A2: To find the first 5 terms of the sequence, we need to substitute the values of n into the sequence formula and calculate the corresponding terms. * For n = 1, we have $$t_1 = 1 + \frac{(-1)^1}{2(1)^2} = 1 - \frac{1}{2} = \frac{1}{2}$$. * For n = 2, we have $$t_2 = 1 + \frac{(-1)^2}{2(2)^2} = 1 + \frac{1}{8} = \frac{9}{8}$$. * For n = 3, we have $$t_3 = 1 + \frac{(-1)^3}{2(3)^2} = 1 - \frac{1}{18} = \frac{17}{18}$$. * For n = 4, we have $$t_4 = 1 + \frac{(-1)^4}{2(4)^2} = 1 + \frac{1}{32} = \frac{33}{32}$$. * For n = 5, we have $$t_5 = 1 + \frac{(-1)^5}{2(5)^2} = 1 - \frac{1}{50} = \frac{49}{50}$$. **Q3: What is the value of f(3) given the function $$f(x) = 5x + 2$$?** A3: To find the value of f(3), we need to substitute x = 3 into the function formula and calculate the corresponding value. $$f(3) = 5(3) + 2 = 15 + 2 = 17$$. **Q4: What is the value of f(-2) given the function $$f(x) = 5x + 2$$?** A4: To find the value of f(-2), we need to substitute x = -2 into the function formula and calculate the corresponding value. $$f(-2) = 5(-2) + 2 = -10 + 2 = -8$$. **Q5: What is the value of f(0) given the function $$f(x) = 5x + 2$$?** A5: To find the value of f(0), we need to substitute x = 0 into the function formula and calculate the corresponding value. $$f(0) = 5(0) + 2 = 0 + 2 = 2$$. **Q6: What is the value of f(a) given the function $$f(x) = 5x + 2$$?** A6: To find the value of f(a), we need to substitute x = a into the function formula and calculate the corresponding value. $$f(a) = 5a + 2$$. **Q7: What is the value of f(x + 1) given the function $$f(x) = 5x + 2$$?** A7: To find the value of f(x + 1), we need to substitute x = x + 1 into the function formula and calculate the corresponding value. $$f(x + 1) = 5(x + 1) + 2 = 5x + 5 + 2 = 5x + 7$$. **Q8: What is the value of f(x - 2) given the function $$f(x) = 5x + 2$$?** A8: To find the value of f(x - 2), we need to substitute x = x - 2 into the function formula and calculate the corresponding value. $$f(x - 2) = 5(x - 2) + 2 = 5x - 10 + 2 = 5x - 8$$. **Q9: What is the value of f(2x) given the function $$f(x) = 5x + 2$$?** A9: To find the value of f(2x), we need to substitute x = 2x into the function formula and calculate the corresponding value. $$f(2x) = 5(2x) + 2 = 10x + 2$$. **Q10: What is the value of f(3x - 1) given the function $$f(x) = 5x + 2$$?** A10: To find the value of f(3x - 1), we need to substitute x = 3x - 1 into the function formula and calculate the corresponding value. $$f(3x - 1) = 5(3x - 1) + 2 = 15x - 5 + 2 = 15x - 3$$. **Q11: What is the value of f(x/2) given the function $$f(x) = 5x + 2$$?** A11: To find the value of f(x/2), we need to substitute x = x/2 into the function formula and calculate the corresponding value. $$f(x/2) = 5(x/2) + 2 = 2.5x + 2$$. **Q12: What is the value of f(2x + 1) given the function $$f(x) = 5x + 2$$?** A12: To find the value of f(2x + 1), we need to substitute x = 2x + 1 into the function formula and calculate the corresponding value. $$f(2x + 1) = 5(2x + 1) + 2 = 10x + 5 + 2 = 10x + 7$$. **Q13: What is the value of f(3x + 2) given the function $$f(x) = 5x + 2$$?** A13: To find the value of f(3x + 2), we need to substitute x = 3x + 2 into the function formula and calculate the corresponding value. $$f(3x + 2) = 5(3x + 2) + 2 = 15x + 10 + 2 = 15x + 12$$. **Q14: What is the value of f(x - 1/2) given the function $$f(x) = 5x + 2$$?** A14: To find the value of f(x - 1/2), we need to substitute x = x - 1/2 into the function formula and calculate the corresponding value. $$f(x - 1/2) = 5(x - 1/2) + 2 = 5x - 5/2 + 2 = 5x - 1/2$$. **Q15: What is the value of f(2x - 1) given the function $$f(x) = 5x + 2$$?** A15: To find the value of f(2x - 1), we need to substitute x = 2x - 1 into the function formula and calculate the corresponding value. $$f(2x - 1) = 5(2x - 1) + 2 = 10x - 5 + 2 = 10x - 3$$. **Q16: What is the value of f(x/3) given the function $$f(x) = 5x + 2$$?** A16: To find the value of f(x/3), we need to substitute x = x/3 into the function formula and calculate the corresponding value. $$f(x/3) = 5(x/3) + 2 = 5/3x + 2$$. **Q17: What is the value of f(3x + 1) given the function $$f(x) = 5x + 2$$?** A17: To find the value of f(3x + 1), we need to substitute x = 3x + 1 into the function formula and calculate the corresponding value. $$f(3x + 1) = 5(3x + 1) + 2 = 15x + 5 + 2 = 15x + 7$$. **Q18: What is the value of f(x - 3/2) given the function $$f(x) = 5x + 2$$?** A18: To find the value of f(x - 3/2), we need to substitute x = x - 3/2 into the function formula and calculate the corresponding value. $$f(x - 3/2) = 5(x - 3/2) + 2 = 5x - 15/2 + 2 = 5x - 7/2$$. **Q19: What is the value of f(2x + 1) given the function $$f(x) = 5x + 2$$?** A19: To find the value of f(2x + 1), we need to substitute x =