Consider the sequence : $2,6,18,54, \ldots$ First term $=2$ Second term $=2 \times 3=6$ Third term $=6 \times 3=18$ Fourth term $=18 \times 3=54$ and so on. Sequences starting with a non-zero number, and each succeeding term got by multiplying the preceding term by a fixed number except zero, are called geometric sequences. The fixed number multiplied is the common ratio of the sequence. Thus, in the geometric sequence $2,6,18,54, \ldots$ the first term is 2 and the common ratio is 3 . a The first term of a geometric sequence is 3 and common ratio is 2 . Find its second and third terms. b Which of the following is a geometric sequence? i $2,4,6,8, \ldots$ ii $2,4,8,16, \ldots$. iii $1,4,9,16, \ldots$ c What is the common ratio of the geometric sequence $5,20,80,320, \ldots$. d Write the next term of the geometric sequence $3,9,27, \ldots$

Explanation

To find the second and third terms of the geometric sequence with first term 3 and common ratio 2, we can multiply the first term by the common ratio to get the second term, and then multiply the second term by the common ratio to get the third term. The second term is 3 * 2 = 6, and the third term is 6 * 2 = 12. A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a fixed number, except zero. In the sequence 2, 4, 6, 8, ..., each term is obtained by adding 2 to the previous term, so it is not a geometric sequence. In the sequence 2, 4, 8, 16, ..., each term is obtained by multiplying the previous term by 2, so it is a geometric sequence. In the sequence 1, 4, 9, 16, ..., each term is obtained by squaring the previous term, so it is a geometric sequence. The common ratio of the geometric sequence 5, 20, 80, 320, ... is 4, since each term is obtained by multiplying the previous term by 4. The next term of the geometric sequence 3, 9, 27, ... is 81, since each term is obtained by multiplying the previous term by 3.

Explanation

⬆ Related Topic

📘 Syllabus

📝 Practice Questions

🤖 Practice with AI

📚 Related Concepts