1Consider the sequences 2, 4, 8, 16, 32, ... and 81, 27, 9, 3, ... How are the numbers arranged in both sequences?
2Is the difference between a term and its preceding term constant in both sequences 2, 4, 8, 16, 32, ... and 81, 27, 9, 3, ...?
3What relation do you see in any two consecutive terms of each of the sequences 2, 4, 8, 16, 32, ... and 81, 27, 9, 3, ...?
4What characteristics are found in both sequences 2, 4, 8, 16, 32, ... and 81, 27, 9, 3, ...?
5Define geometric sequence with an example.
6If a, ar, \( ar^2 \), ... are the first, second and third terms of a geometric sequence, write the sixth term.
7Which of the following are geometric sequences? -1, 6, -36, 316, ...; -1, 1, 4, 8, ...; 4, 16, 36, 64, ...; -3, -15, -75, -375, ...; -2, -4, -8, -16, ...; 1, -5, 25, -125, ...
8Find the common ratio and the next two terms of each geometric sequence: 2, 1, \( \frac{1}{2} \), ...; 5, -10, 20, ...; 4, 12, 36, ...; a, ab, \( ab^2 \), ...
9Find the first term of a geometric sequence whose sixth term is 729 and whose common ratio is 3.
10Determine the first term of a geometric sequence whose 4th term is -8 and whose common ratio is \( -\frac{1}{2} \).
11If the first term of a GP is 4 and the 5th term is 64, find the common ratio.
12How many terms are there in the geometric sequence 3, 12, 48, ..., 192?
13Find the number of terms in the sequence 1, 5, 25, ..., 3125.
14Find the value of x if x, x + 4 and x + 6 are consecutive terms of a geometric sequence.
15Find the value of p if 2p, 2p + 3 and 2p + 9 are three consecutive terms of a geometric sequence. Also find the common ratio.
16The third term of a geometric progression is -108 and the sixth term is 32. Find the common ratio and the first term, the geometric sequence, and the 10th term.
17The second and fifth terms of a geometric sequence are 750 and -6. Find the common ratio and the first term, the geometric sequence, and the 8th term.
18The fourth term of a GP is the square of its second term, and the first term is -3. Determine the 8th term of the GP.
19In a geometric sequence, the 2nd term is 9, and 27 times the 8th term is equal to the 5th term. Find the 12th term.
20The first term of a geometric sequence is 1. The ninth term exceeds the fifth term by 240. Find the possible values of the eighth term.
21The first three terms of a geometric sequence are k + 4, k and 2k - 15, where k is a positive constant. Show that k = 12, find the common ratio and find the 10th term.
22A family saves Rs. 2,000 in the first week of the new year. If they double the amount they save every week after that, how much will they save in the 2nd, 3rd, 4th and 5th weeks? Complete a table of week number and amount saved, and explain your reason.
23Why is the power of r in the general term \( n - 1 \) and not n?
24Find the 8th term of the sequence 1, -3, 9, -27, ...
25In a GP the 2nd term is 6 and the 5th term is 162. Find the first term and the 7th term.
26Is 7, 7, 7, 7, ... a geometric sequence? Explain.