7 Directions: Read and underst What pattern is neces The arithmetic sequer Given the arithmetic sequence Find the 10th term of an arithmet If the first term of a geometric eq Calculate the common ratio and sequence: 3, 6, 12, 24, ST-GOO If three geometric means are inserted between If three arithmetic means are inserted between B Find the nth term if the sequence 7, 11, 15. 10. Find the nth term if the sequence 5, 25,
Answer (1)
Answer:Here's a step-by-step guide to answer the questions based on arithmetic and geometric sequences: 1. What pattern is necessary in an arithmetic sequence? In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference (d). 2. Given the arithmetic sequence: 3, 6, 12, 24, ... Note: This sequence is not arithmetic because the difference between terms is not constant. - 6 - 3 = 3- 12 - 6 = 6- 24 - 12 = 12 (differences keep increasing) This is actually a geometric sequence where each term is multiplied by 2 to get the next term. 3. Find the 10th term of an arithmetic sequence The formula for the nth term of an arithmetic sequence is: a_n = a_1 + (n - 1)d wherea_1 = first term,d = common difference, andn = term number. 4. If the first term of a geometric sequence is given, calculate the common ratio and the sequence The formula for the nth term of a geometric sequence is: a_n = a_1 \times r^{n-1} wherea_1 = first term,r = common ratio. To find the common ratio r, divide the second term by the first term: r = \frac{a_2}{a_1} 5. If three geometric means are inserted between two terms This means 5 terms in total: the first term, three geometric means, and the last term. Find the common ratio using: r = \left(\frac{a_{n}}{a_1}\right)^{\frac{1}{n}} where n = total number of intervals (number of terms - 1) 6. If three arithmetic means are inserted between two numbers Similarly, for arithmetic means, there will be 5 terms total. Find the common difference d by: d = \frac{a_n - a_1}{n} where n = number of intervals (terms - 1) 7. Find the nth term if the sequence is: 7, 11, 15, ... This is an arithmetic sequence with: - First term a_1 = 7- Common difference d = 11 - 7 = 4 The formula is: a_n = 7 + (n - 1) \times 4 = 4n + 3 8. Find the nth term if the sequence is: 5, 25, ... This is likely a geometric sequence. - First term a_1 = 5- Common ratio r = \frac{25}{5} = 5 The nth term formula: a_n = 5 \times 5^{n - 1} = 5^n
